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Texts and Readings in Mathematics 74

Theory of Semigroups and Applications

Kalyan SinhaSachi Srivastava

Texts and Readings in Mathematics

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Kalyan B. Sinha • Sachi Srivastava

Theory of Semigroupsand Applications

123

Kalyan B. SinhaJawaharlal Nehru Centre for AdvancedScientific Research

BangaloreIndia

Sachi SrivastavaDepartment of MathematicsUniversity of DelhiNew DelhiIndia

ISSN 2366-8725 (electronic)Texts and Readings in MathematicsISBN 978-981-10-4864-7 (eBook)DOI 10.1007/978-981-10-4864-7

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About the Authors

Kalyan B. Sinha is professor and the SERB-fellow at the Jawaharlal Nehru Centrefor Advanced Scientific Research (JNCASR), and at the Indian Institute of Science(IISc), Bengaluru. Professor Sinha is an Indian mathematician who specialised inmathematical theory of scattering, spectral theory of Schrödinger operators, andquantum stochastic processes. He was awarded in 1988 the Shanti SwarupBhatnagar Prize for Science and Technology, the highest science award in India, inthe mathematical sciences category. A Fellow of the Indian Academy of Science(IASc), Bengaluru, Indian National Science Academy (INSA), New Delhi, and TheWorld Academy of Sciences (TWAS), Italy, he completed his PhD from theUniversity of Rochester, New York, U.S.A.

Sachi Srivastava is associate professor at the Department of Mathematics,University of Delhi, India. She obtained her DPhil degree from Oxford University,UK and the MTech degree from the University of Delhi, India. Her areas of interestare functional analysis, operator theory, abstract differential equations, operatoralgebras. She is also a life member of the American Mathematical Society andRamanujan Mathematical Society.

v

Preface

Semigroups (or groups, in many situations) of maps or operators

in a linear space have played important roles, mathematically en-

capsulating the idea of homogeneous evolution of many observed

systems, physical or otherwise. As an abstract mathematical disci-

pline, the theory of semigroups is fairly old, with the classical text,

Functional Analysis and Semigroups by Hille and Phillips [12] be-

ing probably the first one of its kind. Indeed, there have been a

good number of books and monographs on this topic written over

the years, many of which have been referred to in the present text.

Perhaps one of the reasons for having so many texts in this one

area of advanced mathematical analysis is the fact that the basic

theory of semigroups finds many applications in numerous areas of

enquiry: partial and ordinary differential equations, the theory of

probability and quantum and classical mechanics to name just a

few. In the present endeavour, along with the systematic develop-

ment of the subject, there is an emphasis on the explorations of

the contact areas and interfaces, supported by the presentations of

explicit computations, wherever feasible.

This book is aimed at the students in the masters level as well

as those in a doctoral programme in universities and research insti-

tutions and envisages the pre-requisites as: (i) a good understanding

viii Preface

of real analysis with elements of the theory of measures and inte-

gration, (for example as in [23]), (ii) a first course in functional

analysis and in the theory of operators, say as in [5]. Many exam-

ples have been given in each chapter, partly to initiate and motivate

the theory developed and partly to underscore the applications. As

mentioned earlier, several of these involve detailed analytical com-

putations, many of which have been undertaken in the text and

some others left as exercises. Instead of making a separate section

on exercises, they appear in line, in bold and in the relevant places

as the subject develops and the readers are encouraged to solve as

many of them as possible. It is suggested that a beginner may read

chapters 1 through 4 (except for sections 3.3 and 3.4) and leave the

rest for a second reading. In the Appendix we have collected some

standard results from the theory of unbounded operators, Fourier

transforms and Sobolev spaces which are required in our treatment

of the subject. It is worthwhile to bring to the attention of the

reader the fact that we have used the notation 〈·, ·〉 to denote the

inner product in Hilbert spaces as well as to represent dual pair-

ing, and 〈·, ·〉 will be taken to be linear in the left and conjugate

linear in the right entry. The present text arose out of the notes

of the lectures given by the first author (K. B. S.) – twice at the

Delhi Centre of the Indian Statistical Institute and once at the In-

dian Institute of Science, Bangalore and the interaction with the

students of those courses has helped shape the final product. Of

course, many existing texts on the subject have influenced the au-

thors and a particular mention needs to be made of the classical

treatise [12] and the books [11], [15] [19] and [27]. The monographs

[2] and [8] have also been referred to frequently. The authors re-

gret that the bibliography is far from exhaustive, instead they were

guided only by the need of the topics treated.

Preface ix

The choice of topics in this vastly developed subject is a diffi-

cult one and the authors have made an effort to stay closer to ap-

plications instead of bringing in too many abstract concepts. While

the chapters 2 and 3 make up the fundamentals of any discourse on

semigroup theory, the first chapter contains background material,

some of which are also of independent interest. Chapter 4 deals with

the issue of the stability of classes of semigroups under small per-

turbations as well as the generalized strong continuity of semigroups

with respect to a parametric dependence. The chapters 5 and 6 deal

with special material, opening avenues for many applications: the

remarkable theorem of Chernoff leading to the Trotter-Kato prod-

uct formula which in turn motivates the Feynman-Kac formula for

a Schrodinger semigroup, and the Central Limit Theorem. Chap-

ter 6 deals with positivity-preserving (or semi-Markov) semigroups,

having its origin in the theory of probability and considers perturba-

tions, not small in the sense of Chapter 4. The motivation for some

of the material in Chapter 5 and Chapter 6 comes from the theory

of probability and for an introduction to elements of that subject,

the reader may consult [18]. The last chapter gives a glimpse of how

the tools of the semigroup theory can be used to understand par-

tial differential operators in particular the wave and Schrodinger

operators.

The first author (K. B. S.) thanks the Indian Statistical In-

stitute, the Indian Institute of Science and most importantly the

Jawaharlal Nehru Centre for Advanced Scientific Research, Ban-

galore, for ready assistance, both direct and indirect, in making

this project a reality. He has special words of gratitude for the De-

partment of Science and Technology, Government of India, for the

SERB-Distinguished Fellowship, and for his wife Akhila for infinite

patience. The second author (S. S.) would like to acknowledge the

Preface

support of the Department of Mathematics, University of Delhi in

this endeavour and of her husband, Manik. It is also a pleasure

to thank Tarachand Prajapati of the Department of Mathematics

at the University of Delhi for help, particularly with regards to the

drawing of the figure in the book. Last but not the least, the authors

are grateful to the anonymous reviewer for many helpful comments

for the improvement in the presentation.

Kalyan B. Sinha Sachi Srivastava

Jawaharlal Nehru Centre for Department of Mathematics

Advanced Scientific Research University of Delhi

Bangalore Delhi

October 2016

x

Contents

Preface vii

1 Vector-valued functions 1

1.1 Vector-valued functions . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Measurability implies continuity . . . . . . . . . . . . . . . . 10

1.4 Operator valued functions . . . . . . . . . . . . . . . . . . . . 12

1.5 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 C0-semigroups 21

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 The generator . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 The Hille-Yosida Theorem . . . . . . . . . . . . . . . . . . . 32

2.4 Adjoint semigroups . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Examples of C0-semigroups and their generators . . . . . . . 39

3 Dissipative operators and holomorphic semigroups 53

3.1 Dissipative operators . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Stone’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 Holomorphic semigroups . . . . . . . . . . . . . . . . . . . . 66

3.4 Some examples of holomorphic semigroups . . . . . . . . . . 77

4 Perturbation and convergence of semigroups 81

4.1 Perturbation of the generator of a C0-semigroup . . . . . . . 81

4.2 Relative boundedness and some consequences . . . . . . . . . 86

4.3 Convergence of semigroups . . . . . . . . . . . . . . . . . . . 90

xii Contents

5 Chernoff’s Theorem and its applications 97

5.1 Chernoff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 97

5.2 Applications of Trotter-Kato and Chernoff Theorem . . . . . 100

6 Markov semigroups 115

6.1 Probability and Markov semigroups . . . . . . . . . . . . . . 115

6.2 Construction of Markov semigroups on a discrete state space 118

7 Applications to partial differential equations 137

7.1 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . 138

7.2 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . 141

7.3 Schrodinger equation . . . . . . . . . . . . . . . . . . . . . . 144

Appendix 147

A.1 Unbounded operators . . . . . . . . . . . . . . . . . . . . . . 147

A.2 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . 153

A.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 158

References 161

Index 165

Chapter 1

Vector-valued functions

This chapter is mostly of a preliminary nature. In the first section we collect

results on measurability and integrability of vector-valued functions that will

be useful throughout. For a more comprehensive treatment of these concepts

in a general setting, we refer the reader to [2] and [12]. The second section

introduces the Bochner integral. The connection between measurability and

continuity of subadditive functions is dealt with in Section 3 while operator

valued functions and general one-parameter semigroups on Banach spaces are

introduced in sections 4 and 5 respectively.

1.1 Vector-valued functions

This section introduces various notions of measurability of vector-valued func-

tions and the connections between them. We assume that the reader is familiar

with the basics of the theory of measure and integration for scalar valued

functions. In the sequel, X shall denote a Banach space and B(X) the space

of bounded, linear operators on X, (Ω,�, μ) will be a σ-finite measure space

while χ� shall denote the indicator function of the set �.

Definition 1.1.1. Let (Ω,�, μ) be a σ-finite measure space and consider

fn, f : Ω→ X, where n ∈ N. The sequence {fn} is said to converge to f

1. almost everywhere if there exists a μ-null set E0 ∈ � such that given

ε > 0, for each t /∈ E0, there is an nε,E0 ∈ N such that

‖fn(t)− f(t)‖ < ε ∀n ≥ nε,E0 ;

1© Springer Nature Singapore Pte Ltd. 2017 and Hindustan Book Agency 2017K.B. Sinha and S. Srivastava, Theory of Semigroups and Applications,Texts and Readings in Mathematics, DOI 10.1007/978-981-10-4864-7_1

2 Vector-valued functions

2. uniformly almost everywhere in Ω if for every ε > 0 there exists a set

Eε ∈ � with μ(Eε) < ε and for every δ > 0, there exists an integer nδ,ε

such that

‖fn(t)− f(t)‖ < δ for all t /∈ Eε and for all n > nδ,ε;

3. in measure if for every ε > 0,

μ{t ∈ Ω: ‖fn(t)− f(t)‖ > ε

}→ 0 as n→∞.

For the case X = C, it is clear that (2)⇒ (1) and (3). If μ(Ω) <∞, then

(1)⇒ (2) and (3). Further, if (3) holds, then there exists a subsequence {fnk}

converging almost everywhere to f.

Definition 1.1.2. A function f : Ω→ X is said to be

1. finitely valued if there exists a sequence {�k : 1 ≤ k ≤ n} for some n ∈ N

of mutually disjoint measurable subsets of Ω and vectors x1, ..., xn in X

such that

f(t) =

⎧⎨⎩∑n

k=1 xkχ�k(t), if t ∈ � = ∪�k,

0 if t /∈ �;

2. simple if it is finitely valued and μ(�k) <∞;

3. countably valued if there exists a sequence {�k : k ∈ N} of mutually

disjoint measurable subsets of Ω and vectors x1, x2, ... in X such that

f(t) =

⎧⎨⎩∑∞

n=1 xnχ�n(t), if t ∈ � = ∪n�n,

0 if t /∈ �;

4. separably valued if f(Ω) is separable in X ;

5. almost separably valued if there exists a μ-null set E ∈ � such that

f(Ω \ E) is separable;

6. weakly measurable if t �→ F (f(t)) is measurable for all F ∈ X∗, the dual

of X ; and

7. strongly measurable if there exists a sequence of countably valued func-

tions converging almost everywhere to f .

1.1. Vector-valued functions 3

It is not difficult to see that if f, g : Ω → X are strongly measurable,

then so is the function f + g. Further, if h : X → Y is continuous, Y being

any Banach space, then h ◦ f is strongly measurable if f is. In particular, this

implies that the function t→ ‖f(t)‖ from Ω to R is measurable.

A subset Λ of X∗ is determining for X if

‖x‖ = sup{|x∗(x)| : x∗ ∈ Λ} for all x ∈ X.

Clearly, ‖x∗‖ ≤ 1 for all x∗ ∈ Λ.

Lemma 1.1.3. If X is separable, then both X and X∗ have countable deter-

mining sets.

Proof. Let {yn} ∈ X be a countable dense set in X . By the Hahn Banach

theorem, there exists a countable subset Λ = {x∗n : n ∈ N} of X∗ such that

x∗n(yn) = ‖yn‖ with ‖x∗

n‖ = 1. For any x ∈ X and ε > 0, there exists n0 ∈ N

such that

ε ≥ ‖x− yn0‖ > |‖x‖ − ‖yn0‖|and

‖x‖ ≥ |x∗n0(x)| = ∣∣x∗

n0(yn0) + x∗

n0(x− yn0)

∣∣ ≥ ‖yn0‖ − ε ≥ ‖x‖ − 2ε.

This implies that Λ is determining for X .

Next let Λ0 be a countable dense set in the unit sphere S1(X) of X. Then,

for x∗ ∈ X∗, we have that

sup{|x∗(xn)| : xn ∈ Λ0

}= sup

{|x∗(x)|, ‖x‖ ≤ 1}= ‖x∗‖.

Thus Λ0 considered as a subset of X∗∗ via the canonical embedding of X in

X∗∗ gives a countable determining set for X∗.

�

Theorem 1.1.4. If f : Ω→ X is weakly measurable and if there exists a count-

able determining set Λ for X, then t �→ ‖f(t)‖ is measurable.

Proof. Suppose Λ = {x∗n : n ∈ N}. Since

‖f(t)‖ = supn

∣∣x∗n(f(t))

∣∣and t �→ x∗

n(f(t)) is measurable for each n ∈ N, the conclusion follows.

�


The next theorem gives a very useful criterion for strong measurability of

a vector-valued function, involving measurability of scalar functions, which is

easier to check.

Theorem 1.1.5. An X-valued function is strongly measurable if and only if it

is weakly measurable and almost separably valued.

Proof. Suppose first that f : Ω→ X is strongly measurable. Then there exists

a null set E ∈ � and a sequence {fn} of countably valued functions such that

‖fn(t)− f(t)‖ → 0 for all t /∈ E.

Therefore, for any x∗ ∈ X∗, x∗(f(t) − fn(t)) → 0 for all t /∈ E. Now t �→x∗(fn(t)) is a countably C-valued function and hence measurable. This implies

that t �→ x∗(f(t)) is also measurable. Thus f is weakly measurable. Further,

the closed linear span M of the countable subset {fn(Ω) : n ∈ N} of X is

a separable subspace of X. Clearly, f(Ω \ E) ⊂ M. Therefore, f is almost

separably valued.

Conversely, assume that f is almost separably valued and weakly mea-

surable. Let E ∈ � be a μ-null set such that f(Ω \ E) is separable. Replacing

X by the closed linear span of {f(Ω \ E)} if required, we can assume that X

itself is separable. By Lemma 1.1.3 there exists a countable determining set Λ

for X and it follows from Theorem 1.1.4 that t �→ ‖f(t)‖ is measurable. Let

Ω0 = {t ∈ Ω \ E : ‖f(t)‖ > 0}. Then, Ω0 ∈ � and the function t �→ f(t) − x0

is weakly measurable on Ω0 for each x0 ∈ X. Therefore, by Theorem 1.1.4,

t �→ ‖f(t)− x0‖ is measurable on Ω0. Since f(Ω \ E) is separable, there exists

a countable dense subset, say {fn : n ∈ N} in f(Ω \ E). Let ε > 0. Set

En = {t ∈ Ω0 : ‖f(t)− fn‖ < ε}.

Then En is measurable and since {fn} is dense in f(Ω \ E),∪En = Ω0. Set

Fn = En \ ∪k<nEk. Then, {Fn} is a sequence of pairwise disjoint, measurable

sets such that ∪Fn = Ω0. Define, for each ε > 0,

gε(t) =

{fn for all t ∈ Fn,

0 for all t /∈ Ω0.

Clearly, each gε is countably valued and

‖f(t)− gε(t)‖ ={‖f(t)− fn‖ if t ∈ Ω0

‖f(t)‖ = 0 if t /∈ Ω0.

1.2. The Bochner integral 5

Therefore, f is the limit of countably valued functions gε, uniformly with respect

to t ∈ Ω \ E, and hence is strongly measurable. �

Corollary 1.1.6. If X is separable, then weak and strong measurability are

equivalent.

1.2 The Bochner integral

Once we have a notion of measurable vector-valued functions, it is natural to ask

when would such a function be integrable or rather, what is meant by an integral

of a function in this context. For our purpose it is enough, while answering

this question, to restrict attention to functions defined on the real line or its

subintervals. Thus, from now on, we shall assume Ω to be (0,∞) or R+ = [0,∞)

or R or a finite subinterval I of R and that μ is the Lebesgue measure on Ω,

denoted by m. While several extensions of the Riemann and Lebesgue integrals

to vector-valued functions exist, here we discuss in detail only the Bochner

Integral – a generalisation or extension of the Lebesgue integral to the vector-

valued case. It is to be noted that subsequently, while writing the integral of

a function with respect to a measure we shall not, in general, mention the

underlying measure explicitly and for integrals with Lebesgue measure, the

traditional “dt” shall be used. This section ends with Lemma 1.2.5, in which

the definition of the Riemann integral of vector-valued functions and a few of

its useful properties are given.

For a simple function f : Ω→ X of the form f(t) =∑n

k=1 xkχ�k(t), where

x1, . . . , xn ∈ X, χ� is the real valued indicator (or characteristic) function of

the measurable set �, and m(�k) < ∞, ∀k = 1, . . . , n, n ∈ N, we define its

integral by

∫Ω

f(t) dt :=

n∑k=1

xkm(�k). (1.1)

Definition 1.2.1. A (strongly) measurable function f is called Bochner inte-

grable if there exists a sequence {fn}n∈N of simple functions on Ω such that

fn → f pointwise almost everywhere and

limn→∞

∫Ω

‖fn(t)− f(t)‖ dt = 0. (1.2)


Further, if f is Bochner integrable, then its Bochner integral is defined to

be ∫Ω

f(t) dt := limn

∫Ω

fn(t) dt. (1.3)

That the limit in (1.3) above exists is a consequence of the fact that (1.2) forces

the sequence{ ∫

Ω fn(t) dt}to be Cauchy.

This limit is independent of the choice of the sequence fn. Indeed, suppose

{fn} and {gn} are two sequences satisfying (1.2) for the given f , and set

a := limn

∫Ω

fn(t) dt and b := limn

∫Ω

gn(t) dt.

Let r2n(t) = fn(t), r2n−1(t) = gn(t). Then we see that the sequence {rn} also

satisfies the conditions of Definition (1.2.1), so that limn

∫Ω rn(t) dt exists. Since

a, b are limit points of the convergent sequence∫Ωrn(t) dt, it follows that a = b.

In fact, in the above definition, we can choose fn to be step functions,

that is, simple functions for which the sets �k are disjoint intervals of R. Note

that if X = C, then the Bochner integral of f is nothing but the Lebesgue

integral.

Lemma 1.2.2. If f : Ω→ X is strongly measurable, then

(a) f is the limit of a sequence of countably valued functions in ess sup norm,

that is, ess supt∈Ω ‖f(t)− fn(t)‖ → 0, and

(b) f is the pointwise limit almost everywhere of a sequence of simple func-

tions.

Proof. Let f be strongly measurable. Then, by Theorem 1.1.5 it is weakly mea-

surable and almost separably valued. Let ε > 0. Then following the construction

as in the proof of the second part of Theorem 1.1.5, define gε :=∑∞

n=1 fnχFn,ε ,

where Fn,ε = Fn and we have added ε to stress the dependence on ε. Let t ∈ Ω.

If t /∈ Ω0 ∪ E then f(t) = 0 = gε(t). If t ∈ Ω0, then there exists n ∈ N, such

that t ∈ Fn,ε. Therefore, ‖f(t)− gε(t)‖ < ε for all t ∈ Ω\E. Since f is the limit

of countably valued functions gε, uniformly with respect to t ∈ Ω \E, the first

part is proved.

For (b), let Ω = ∪nIn, where each In is an increasing sequence of bounded

subintervals of Ω. For each n, let Jn := In ∩(∪cn

n=1 Fn,3−n

), where cn is chosen


so that m(In \ Jn) < 3−n. Set hn := g3−nχJn , n ∈ N. If t ∈ ∩∞n=kJn for some

k ≥ 1, then

‖f(t)− hn(t)‖ = ‖f(t)− g3−n(t)‖ < 3−n

for all n ≥ k. Thus limn→∞ hn(t) = f(t) for all t ∈ J = ∪∞k=1∩∞

n=kJn.Moreover,

for k ≤ j,

m(Ij \ ∩∞

n=kJn) ≤ ∞∑

n=k

m(In \ Jn) < 3−k.

Thus Ij \ J is null for each j. Therefore, limn→∞ hn(t) = f(t), for almost all

t ∈ Ω. �

The class of Bochner integrable functions has a very nice characterisation,

making them relatively easy to use.

Theorem 1.2.3. A function f : Ω→ X is Bochner integrable if and only if it is

strongly measurable and the function t �→ ‖f(t)‖ is Lebesgue integrable. If f

is Bochner integrable, then∥∥ ∫Ω

f(t) dt∥∥ ≤ ∫

Ω

‖f(t)‖ dt. (1.4)

Proof. Suppose first that f is Bochner integrable. Then, there exists a sequence

{fn} of simple functions which approximate f in the sense of Definition 1.2.1.

Thus, f is strongly measurable and the function t �→ ‖f(t)‖ is measurable.

Since ∫Ω

‖f(t)‖ dt ≤∫Ω

‖f(t)− fn(t)‖ dt+∫Ω

‖fn(t)‖ dtand since

limn→∞

∫Ω

‖fn(t)− f(t)‖ dt = 0,

it follows that t �→ ‖f(t)‖ is integrable. Further,∥∥ ∫Ω

f(t) dt∥∥ = lim

n

∥∥ ∫Ω

fn(t) dt∥∥

≤ limn→∞

∫Ω

‖fn(t)‖ dt =∫Ω

‖f(t)‖ dt.

Conversely, suppose that f is strongly measurable. Then by Lemma 1.2.2

(b), there is a sequence {gn} of simple (or finitely valued ) functions converging

pointwise to f on a subset Ω \ Ω0 of Ω, and m(Ω0) = 0. Set for t ∈ Ω \ Ω0,

fn(t) :=

⎧⎨⎩gn(t) if ‖gn(t)‖ ≤ 2‖f(t)‖(1 + n−1),

0, otherwise.


Note that if for t ∈ Ω \ Ω0, f(t) = 0, then fn(t) = 0 for all n. On the other

hand, if f(t) �= 0, then the set {n ∈ N : ‖gn(t)‖ > 2‖f(t)‖(1 + n−1)} must be

finite. Indeed if this set is infinite, then we can find a subsequence {nk} ⊂ N

such that

‖gnk‖ > 2‖f(t)‖(1 + n−1

k ). (1.5)

Letting k →∞ in (1.5) gives ‖f(t)‖ ≥ 2‖f(t)‖, which is a contradiction. Thus

for sufficiently large n, fn(t) = gn(t) if f(t) �= 0. Therefore, fn converges point-

wise to f on Ω\Ω0. Writing hn(t) = fn(t)−f(t) we see that ‖hn(t)‖ ≤ 5‖f(t)‖and almost everywhere on Ω, limn→∞ hn(t) = 0. Since ‖f‖ is integrable, by the

scalar Dominated Convergence Theorem, we have that limn→∞∫ ‖hn(t)‖ = 0.

This shows that f is Bochner integrable.

�

Lemma 1.2.4. Let f be a bounded, X-valued strongly measurable function on

R+. Then∫ b

a

‖f(t+ δ)− f(t)‖ dt→ 0 as δ → 0, for 0 < a < b <∞.

Proof. Since f is strongly measurable, we may assume without loss of generality

that it is separably valued. Also, it follows from Lemma 1.2.2(a) that there exist

a sequence {fn} of countably valued functions such that

ess supt∈Ω ‖f(t)− fn(t)‖ → 0 as n→∞.

Let ε > 0 be given. Thus there exists an n0 ∈ N, and Ω0 ⊂ Ω, such that

m(Ω0) = 0 and for all n ≥ n0,

supt∈Ω\Ω0

‖f(t)− fn(t)‖ < ε. (1.6)

Fix n = n0. Then for any δ > 0,∫ b

a

‖f(t+ δ)− f(t)‖ dt ≤∫ b

a

‖f(t+ δ)− fn(t+ δ)‖ dt+∫ b

a

‖f(t)− fn(t)‖ dt

+

∫ b

a

‖fn(t+ δ)− fn(t)‖ dt

≤ 2(b− a)ε +

∫ b

a

‖fn(t+ δ)− fn(t)‖ dt. (1.7)


Set

fn(t) =

∞∑k=1

xkχ�k(t). (1.8)

The fact that f is bounded together with (1.6) implies that

ess supt∈Ω\Ω0‖fn(t)‖ ≤M, (1.9)

for some constant M. Using (1.8) and (1.9) we get that sup∞k=1 ‖xk‖ ≤ M1.

Writing �k − δ = {s− δ : s ∈ �k}, we therefore have

∫ b

a

‖fn(t+ δ)− fn(t)‖ dt (1.10)

=

∫ b

a

‖∞∑k=1

xk[χ�k−δ(t)− χ�k(t)]‖ dt

≤∫ b

a

∞∑k=1

‖xk‖∣∣χ�k−δ(t)− χ�k

(t)∣∣ dt

≤ supk‖xk‖

∞∑k=1

m([a, b] ∩ [(�k \ (�k − δ)) ∪ ((�k − δ) \ �k)]

≤M1

∞∑k=1

m([a, b] ∩ [(�k \ (�k − δ)) ∪ ((�k − δ) \ �k)]

→ 0 as δ → 0, (1.11)

by the Dominated Convergence Theorem. Now (1.7) together with (1.11) es-

tablishes the claim.

�

The next lemma defines the vector-valued Riemann integral and collects

a few simple but useful results.

Lemma 1.2.5. (i) Let a, b ∈ R, and let f : [a, b]→ X be a continuous function.

For a partition P = {a = s0 < s1 < s2 < . . . < sn = b} of I let

Ψ(f ;P, a, b) :=∑n

j=1 f(sj)(sj − sj−1) denote the Riemann sum. Then

Ψ(f ;P, a, b) converges, as the partition width |P | = maxnj=1(sj − sj−1)

approaches 0, to an element in X, which shall be called the Riemann

integral of f over the interval [a, b] and written as

∫ b

a

f(s) ds. This f is

also Bochner integrable and the two integrals coincide.


(ii) Let A be a closed operator in X and let f be as in (i) above. Assume that

f(s) ∈ D(A) for every s ∈ [a, b] such that the map s �→ Af(s) from [a, b]

to X is continuous. Then

A

∫ b

a

f(s) ds =

∫ b

a

Af(s) ds. (1.12)

Proof. The proof of (i) is identical to that in the scalar case, using the fact

that the continuity of f implies uniform continuity over I.

Now suppose A and f are as in (ii). Then by (i) both the Riemann

integrals∫ b

a f(s) ds and the integral∫ b

a Af(s) ds exist. Thus the sequence

{Ψ(f, Pn, a, b)} ⊂ D(A) where

Pn = {a < a+ (b− a)/n < a+ 2(b− a)/n < . . . , sn = b},

and converges to∫ b

a f(s) ds while the sequence AΨ(f, Pn, a, b) = Ψ(Af, Pn, a, b)

converges to∫ b

aAf(s) ds as n → ∞. Since A is closed, this implies that∫ b

af(s) ds is in D(A) and (1.12) holds.

�

1.3 Measurability implies continuity

We now explore the relation between measurability and continuity for vector-

valued functions defined on (0,∞). If f : (0,∞)→ X is continuous, then clearly

f is weakly measurable and the countable set {f(t) : t ∈ Q∩ (0,∞)} is dense inthe range of f. Therefore, by Theorem 1.1.5, f is strongly measurable. Thus, as

in the scalar case, continuity implies measurability. The following result shows

that the converse is also true for some special functions. For the purpose of

the next theorem, we define a Banach algebra: An algebra X which is also a

Banach space with respect to a norm ‖ ·‖ such that ‖xy‖ ≤ ‖x‖‖y‖, ∀x, y ∈ X,

is called a Banach algebra. Recall that a function g : (0,∞) → R is said to be

subadditive if g(t+ s) ≤ g(t) + g(s) ∀ t, s ∈ (0,∞).

Lemma 1.3.1. Let X be a real or complex Banach algebra, possibly without a

unit, and let f : (0,∞)→ X be a strongly measurable function satisfying

f(t1)f(t2) = f(t1 + t2), for all t1, t2 ∈ (0,∞).

Then f is bounded in every bounded interval in (0,∞) and is continuous on

(0,∞).

1.3. Measurability implies continuity 11

Proof. Since f is strongly measurable, it follows from Theorem 1.1.5 and The-

orem 1.1.4 that t �→ ‖f(t)‖ is measurable.

Suppose first that f(t) �= 0 for all t ∈ (0,∞). Since for t1, t2 ∈ (0,∞),

‖f(t1 + t2)‖ ≤ ‖f(t1)‖‖f(t2)‖, we have that

log ‖f(t1 + t2)‖ ≤ log ‖f(t1)‖+ log ‖f(t2)‖.

Therefore, the function α : (0,∞)→ R defined by α(t) = log ‖f(t)‖ is subaddi-tive on (0,∞), that is, α(t1+t2) ≤ α(t1)+α(t2) for all t1, t2 ∈ (0,∞). We claim

that α is bounded above on any subinterval (c, d) of (0,∞) where 0 < c < d <

∞. Let a > 0 and α(a) = A. For t+s = a, and t, s > 0, A = α(a) ≤ α(t)+α(s).

If we set

E ={t ∈ (0, a) : α(t) ≥ A

2

},

then

(0, a) = E ∪ (a− E). (1.13)

Indeed, for r ∈ (0, a), if α(r) ≥ A2 , then r ∈ E. Otherwise, α(r) < 2−1α(a), so

that α(a − r) ≥ α(a) − α(r) > A2 . Thus a − r ∈ E whence r ∈ a − E. Now

(1.13) implies that

a ≤ m(E) +m(a− E) = 2m(E) so that m(E) ≥ a

2.

Suppose if possible that α is unbounded in some interval (c, d) where 0 < c <

d <∞. Then there exists a sequence {tn} ⊂ (c, d) such that tn → t0 ≥ c and

α(tn) ≥ 2n, for each n ∈ N.

Therefore, by an argument similar to one used above, we have that for every

n ∈ N, the set

En = {t ∈ (0, d) : α(t) ≥ n}has measure m(En) >

c2 . This implies that the function α takes the value ∞

on a set of measure at least c/2. This is a contradiction. Thus α is bounded

above on (ε, ε−1) for all 1 > ε > 0. Let

α(t) ≤Mε for all t ∈ (ε, ε−1),

and it follows that t �→ ‖f(t)‖ is a bounded measurable function in (ε, ε−1).

Choose a, b, c such that 0 < a < b < c <∞. Then the integral

∫ b

a

f(c−t)f(t) dt


exists as a strong Bochner integral and is equal to f(c)(b − a). Therefore, if

ε > 0,

(b− a)[f(c+ ε)− f(c)] =

∫ b

a

[f(c+ ε− t)− f(c− t)]f(t) dt,

and hence

(b− a)‖f(c+ ε)− f(c)‖ ≤ ∥∥ ∫ c−b

c−a

[f(τ + ε)− f(τ)]f(c− τ) dτ∥∥

≤M

∫ c−b

c−a

‖f(τ + ε)− f(τ)‖ dτ

→ 0, as ε→ 0,

where M = supa≤t≤b ‖f(t)‖, and the convergence to zero is a consequence of

Lemma 1.2.4.

Next we consider the case when there exists a t0 ∈ (0,∞) such that

f(t0) = 0. Then f(t) = 0 for all t ≥ t0. The conclusion of the theorem in this

case is arrived at by following the same proof as before with the open interval

(0,∞) replaced by (0, t0).

�

Remark 1.3.2. Caution: Note that the conclusion of the above Lemma is only

for the open right half line (0,∞) and not for [0,∞).

1.4 Operator valued functions

We now consider the special vector-valued functions which assume values in

B(X), the space of bounded linear operators on some Banach space X. Since

these functions take values in B(X) they are referred to as operator valued

functions. Such functions are of particular relevance to us since a semigroup

of operators on a Banach space X is an operator valued function T : [0,∞)→B(X), satisfying the semigroup property (see (1.5.1) below).

The following definition makes precise the notion of uniform, strong and

weak measurability for operator valued functions.

Definition 1.4.1. An operator valued function T : (Ω,�, μ)→ B(X) is

1.5. Semigroups 13

1. uniformly measurable if there exists a sequence {Tn} of countably (op-

erator) valued functions on Ω converging almost everywhere to T in the

operator norm;

2. strongly measurable if the vector-valued function t �→ T (t)x is strongly

measurable for every x ∈ X ;

3. Weakly measurable if t �→ y∗(T (t)x) is measurable for every x ∈ X and

y∗ ∈ X∗.

Similarly, we may consider continuity for operator valued functions in

various topologies on B(X). However, the uniform, strong and weak forms of

continuity are the ones we will work with most of the time. Therefore, we make

precise the definitions here:

Definition 1.4.2. An operator valued function T on Ω where Ω is either R+ or

I, a finite interval in R, is

1. uniformly continuous if the function t �→ T (t) from Ω to B(X) is contin-

uous with respect to the operator norm;

2. strongly continuous if the vector-valued function t �→ T (t)x from Ω to X

is continuous for every x ∈ X ;

3. weakly continuous if the function t �→ y∗(T (t)x) from Ω to C is continuous

for every x ∈ X and y∗ ∈ X∗.

1.5 Semigroups

In this section we look at semigroups of bounded operators on a Banach space.

As mentioned earlier they may be thought of as operator valued functions with

a particular property.

Definition 1.5.1. A semigroup of operators on the Banach space X is an oper-

ator valued function T : [0,∞)→ B(X) satisfying

T (t)T (s) = T (t+ s) for all s, t ≥ 0. (1.14)

For semigroups, the three types of measurability and continuity we have

defined above are closely connected. Since B(X) is a Banach algebra, a direct


consequence of Lemma 1.3.1 is that a uniformly measurable semigroup is uni-

formly continuous, that is, the map t �→ T (t) from (0,∞) to B(X) is continuous

in the norm topology of B(X).

Theorem 1.5.2. Let T be a uniformly measurable semigroup on a Banach space

X . Then T is uniformly continuous in (0,∞).

In fact, the above result remains true if we replace uniform by strong.

However, the proof of this requires some further work. We first establish the

following lemma.

Lemma 1.5.3. Let T be a semigroup on X which is strongly measurable on

(0,∞). Then the function t �→ ‖T (t)‖ is bounded on [α, β] for all α, β such that

0 < α < β <∞.

Proof. By the Uniform Boundedness Principle, it suffices to show that for every

x ∈ X, the set {T (t)x : α ≤ t ≤ β} is bounded. Suppose that this is not true.

Then for some x ∈ X there exists a c ∈ [α, β] and a sequence(tn) ⊂ [α, β] such

that tn → c as n → ∞ and ‖T (tn)x‖ ≥ n for all n. Strong measurability of T

together with Theorem 1.1.4 and Theorem 1.1.5 implies that t �→ ‖T (t)x‖ is

measurable. An application of Lusin’s Theorem [23, Lusin’s Theorem, page 66]

yields that there exists an M > 0 and a measurable set E ⊂ [0, c] with measure

m(E) >c

2such that

supt∈E

‖T (t)x‖ ≤M.

Now set En = {tn − η : η ∈ E ∩ [0, tn]}. This is a measurable set and for large

enough n, m(En) ≥ c

2. Therefore,

n ≤ ‖T (tn)x‖ ≤ ‖T (tn − η)‖‖T (η)x‖ ≤M‖T (tn − η)‖ (1.15)

Thus, ‖T (t)‖ ≥ nM for all t ∈ En. Denoting lim supn En by F , it follows that

‖T (t)‖ =∞ ∀ t ∈ F. But m(F ) ≥ c2 > 0, implying that T (t) is not defined for

t in a set of strictly positive measure, leading to a contradiction. �

Theorem 1.5.4. Let T be a semigroup on X which is strongly measurable on

(0,∞). Then T is strongly continuous on (0,∞).

Proof. Let x ∈ X and 0 < a < t < b < s. Suppose ε > 0 is so small that

ε < s− t. Using the identity T (s)x = T (t)T (s− t)x, we have

(b− a)[T (s± ε)− T (s)]x =

∫ b

a

T (t)[T (s± ε− t)− T (s− t)]x dt.

1.5. Semigroups 15

From Lemma 1.5.3 it follows that there exists an M > 0 such that ‖T (t)‖ ≤M

for all t ∈ [a, b]. Therefore,

‖(b− a)[T (s± ε)− T (s)]x‖ = ∥∥ ∫ b

a

T (t)[T (s± ε− t)− T (s− t)]x dt∥∥

≤M

∫ b

a

‖[T (s± ε− t)− T (s− t)]x‖ dt

= M

∫ s−a

s−b

‖[T (u± ε)− T (u)]x‖ dt

→ 0 as ε→ 0,

by Lemma 1.2.4. �

Remark 1.5.5. Thus the hypothesis of strong measurability is enough to render

a semigroup into a strongly continuous family on the open interval (0,∞).

However, in general, it may not be possible to extend this continuity to [0,∞).

Those semigroups for which this is valid form the most useful class, viz, C0-

semigroups.

Corollary 1.5.6. Weak one-sided continuity on (0,∞) of a semigroup T on X

implies strong continuity of T on (0,∞).

Proof. Recall that weak one-sided continuity implies weak measurability. For

any a, b with 0 < a < b <∞, and x ∈ X fixed,

the closed linear span of {T (t)x : t ∈ [a, b]}≡ the closed linear span of {T (t)x : t ∈ Q ∩ [a, b]}.

Since every strongly closed linear subspace is weakly closed (as a consequence

of the Hahn-Banach Theorem), it follows therefore that the weakly closed linear

span of {T (t)x : t ∈ Q ∩ [a, b]} is equal to the strongly closed linear span of

{T (t)x : t ∈ Q ∩ [a, b]}.Therefore t �→ T (t)x from [a, b] to X is separably valued. It follows from

Theorem 1.1.5 that this map is strongly measurable and then from Theorem

1.5.4 that it is strongly continuous. Since a, b ∈ (0,∞) are arbitrary, the result

follows. �

The following example shows that strong continuity of a semigroup does

not, in general, imply uniform continuity.


Example 1.5.7. Let X = C0(R+), the Banach space of continuous functions on

R+ which vanish at ∞. Let T be the semigroup defined by setting

(T (t)f)(s) = e−s2tf(s), for all t, s ≥ 0, and for f ∈ X.

Then T is strongly continuous but not uniformly continuous. Indeed, for h, s >

0, t ≥ 0 fixed, f ∈ C0(R+), and ‖f‖ ≤ 1,

T (t+ h)f(s)− T (t)f(s) = (e−s2(t+h) − e−s2t)f(s)

= e−s2t(e−s2h − 1)f(s).

Therefore, ‖T (t+ h)f − T (t)f‖ = sups≥0

|e−s2t(1− e−s2h)f(s)|. (1.16)

Let ε > 0. Then there is a compact set Kε ⊂ [0,∞), such that

|f(s)| < ε/2 ∀s ∈ [0,∞) \Kε.

Since the map t→ e−s2t is uniformly continuous for s in any compact set, there

exists δ ∈ (0, 1] such that

|1− e−s2h| < ε ∀s ∈ Kε, 0 < h < δ.

Then, (1.16) gives

‖T (t)f − T (t+ h)f‖ ≤ max(

sups∈[0,∞)\Kε

|f(s)|, sups∈Kε

|(1 − e−s2h)|) < ε,

for all h such that 0 < h < δ.Thus for each f ∈ C0(R+), ‖T (t)f−T (t+h)f‖ → 0

as h ↓ 0. On the other hand,

‖T (t)− T (t+ h)‖ = sups≥0

|1− e−s2h| = 1 for all h �= 0.

Therefore, T is not uniformly continuous.

However, strong continuity of T does imply that t �→ ‖T (t)x‖ is continuouson (0,∞) for each x ∈ X. Therefore, t �→ ‖T (t)‖ = sup‖x‖=0

‖T (t)x‖‖x‖ is lower

semi-continuous and hence measurable. Consider first the case when T (t) �= 0

for all t ∈ (0,∞). Then the function t �→ log ‖T (t)‖ = α(t) is a measurable

function on (0,∞). Since ‖T (t+s)‖ ≤ ‖T (t)‖‖T (s)‖, t, s ≥ 0, the above function

α is subadditive and by proof of Lemma 1.3.1, is different from +∞ on (0,∞).

In the case that there exists a t0 ∈ (0,∞) such that T (t0) = 0, T (t) = 0 for all

t > t0, and the same conclusion as before may be arrived at by considering α

as a function defined on (0, t0) instead of (0,∞).

1.5. Semigroups 17

Lemma 1.5.8. Let f : (a,∞)→ R, where a ≥ 0, be a subadditive measurable

function. Then

limt→∞

f(t)

t= inf

t>a

f(t)

t<∞.

Proof. Let β = inft>af(t)

t. Then β is either finite or −∞. Suppose first that

β is finite. Let ε > 0. Choose b > a such that f(b) < (β + ε)b and n ∈ N such

that (n+ 2)b ≤ t ≤ (n+ 3)b. Then, for t > a,

β ≤ f(t)

t≤ f(t− nb) + f(nb)

t

≤ nb

t

f(b)

b+

f(t− nb)

t

<nb

t(β + ε) +

f(t− nb)

t. (1.17)

Since t−nb ∈ [2b, 3b], it follows from Lemma 1.3.1 that |f(t−nb)| is bounded.Therefore,

limt→∞

f(t)

t≤ lim

t→∞(nbt(β + ε) +

f(t− nb)

t

)= β + ε.

Thus,

β ≤ limt→∞

f(t)

t≤ β + ε

and since ε > 0 is arbitrary, it follows that

limt→∞

f(t)

t= β.

If β = −∞, then for any m ∈ N we find b ≥ a such thatf(b)

b< −m and

inequality (1.17) shows that for t sufficiently large,f(t)

t< −m. This implies

that limt→∞f(t)

t= −∞.

�

Lemma 1.5.8, when applied to the function f(t) = log ‖T (t)‖, where t ∈(0,∞), gives

w0(T ) := inft>0

log ‖T (t)‖t

= limt→∞

log ‖T (t)‖t

<∞. (1.18)

This w0(T ) is called the type of the semigroup T . It is also referred to as the

exponential growth bound of the semigroup T . The reason for this is apparent

from the next result where some simple properties of the type of a semigroup

are listed.


Theorem 1.5.9. Let T be a semigroup on X and w0 = w0(T ) denote its type.

The following hold:

1. If T is strongly continuous on (0,∞), then w > w0 implies the existence

of Mw > 0 such that

‖T (t)‖ ≤Mwewt for all t > 0.

In fact,

w0 = inf{w ∈ R : there exists Mw ≥ 0 with ‖T (t)‖ ≤Mwewt}.

2. If w0 > −∞, then the spectral radius of T (t) is given by ew0t for each

t ∈ (0,∞).

Proof. Let α = inf{w ∈ R : there exists Mw ≥ 0 with ‖T (t)‖ ≤ Mwewt} and

let w > w0. Since w0 = limt→∞log ‖T (t)‖

t, for ε = w − w0 there exists t0 > 0

such thatlog ‖T (t)‖

t< w0 + ε for all t > t0.

Thus

‖T (t)‖ ≤ e(w0+ε)t = ewt for t > t0. (1.19)

Using Lemma 1.5.3 and (1.19) we can find an Mw such that ‖T (t)‖ ≤ Mwewt

for all t > 0. Thus w ≥ α. Since w > w0 was arbitrary, it follows that w0 ≥ α.

Conversely, let w ∈ R be such that ‖T (t)‖ ≤ Mwewt for all t > 0. Therefore,

for t > 0,log ‖T (t)‖

t≤ logMw

t+ w.

So limt→∞log ‖T (t)‖

t≤ w. This implies that w0 ≤ w so that w0 ≤ α. There-

fore, α = w0.

By the spectral radius formula (see [5]) and definition of type, for t ∈(0,∞),

r(T (t)) = limn→∞

‖T (t)n‖1/n = limn→∞

‖T (nt)‖1/n

= limn→∞

exp( t

ntlog ‖T (nt)‖

)= etw0 .

�

1.5. Semigroups 19

A semigroup T for which ‖T (t)‖ ≤ 1, for all t > 0, that is, the choice

w = 0 and Mw = 1 is permissible, is called a contraction semigroup . Note that

we have not yet assumed strong continuity at t = 0. A semigroup T which is

strongly continuous (equivalently, measurable) on [0,∞) and T (0) = I is called

a C0-semigroup, which will be the subject of discussion in the next chapter.

The following is an example of a contraction C0-semigroup.

Example 1.5.10. Let X = BUC(R+), the space of all bounded uniformly

continuous functions on the half line. Define the semigroup T on X by setting

(T (t)f)(s) = f(t+ s), for all s, t ≥ 0, f ∈ X.

Then T (0) = I and ‖T (t)f‖ = ‖f‖ for all f ∈ X, so that ‖T (t)‖ = 1 for

all t ≥ 0. Thus T is a contraction semigroup with w0(T ) = 0. Further, as a

consequence of uniform continuity, for any f ∈ X,

‖T (t)f − f‖ = sups≥0

‖f(t+ s)− f(s)‖ → 0 as t→ 0.

Therefore, for any fixed s > 0,

‖T (s+ h)f − T (s)f‖ = ‖T (s)(T (h)f − f)‖ ≤ ‖T (s)‖‖T (h)f − f‖ (1.20)

→ 0 as h→ 0.

Thus T is strongly continuous.

Note that the inequality (1.20) actually holds for any semigroup. This

shows that if a semigroup {T (t)}t≥0 is exponentially bounded on [0,∞), then

strong continuity at 0 is sufficient for the semigroup to be strongly continuous

on all of [0,∞). We postpone giving further examples of C0-semigroups and

illustrations of the concept of type until the next chapter, where C0-semigroups

are studied in detail.

Chapter 2

C0-semigroups

In this chapter we concentrate on strongly continuous or more specifically C0-

semigroups of bounded operators on a Banach space. The notion of the gen-

erator of a C0-semigroup is introduced and their properties are dealt with in

detail.

2.1 Introduction

Consider a function T : [0,∞) → Mn(C), satisfying the following properties:

(i) T (0) = I, (ii) T (t + s) = T (t)T (s) ∀ t, s ≥ 0 and (iii) T (t) → I as t → 0.

Then it is not difficult to see that the map t → T (t) is differentiable and

T (t) = eAt for some A ∈ Mn(C) (Exercise 2.1.1). Here continuity, together

with the semigroup property implies differentiability. This implication carries

over to the infinite-dimensional case also, but with a qualification – A may

not be defined everywhere. (Recall that A ∈ Mn(C) may be considered as a

linear operator on Cn, defined everywhere). We prove the above assertion in

this section.

The following simple result will be used repeatedly in the text and is given

for the sake of completeness.

Lemma 2.1.2. Let X be a Banach space and let f ∈ [0, a]→ X be a continuous

function. Then

limt→0+

t−1

∫ t

0

f(s) ds = f(0).


22 C0-semigroups

Proof. Note that the integral exists as a Riemann integral as well as a Bochner

integral (Lemma 1.2.5) and that

[t−1

∫ t

0

f(s) ds− f(0)]= t−1

∫ t

0

[f(s)− f(0)] ds.

Since continuity implies uniform continuity on a compact interval, given ε > 0,

one can find a δ > 0 such that ‖f(s)−f(0)‖ < ε whenever 0 < s < δ. Therefore,

∥∥t−1

∫ t

0

f(s) ds− f(0)∥∥ ≤ t−1

∫ t

0

∥∥f(s)− f(0)∥∥ ds < ε for 0 < t < δ.

�

2.2 The generator

Assume that T : (0,∞) → B(X) is a semigroup of operators and is strongly

continuous on (0,∞). The infinitesimal generator A0 of T is defined in the

following manner. Set

Aηx =T (η)x− x

η, η > 0, and (2.1)

A0x = limη→0+

Aηx, (2.2)

whenever the limit exists. From now on we shall refer to the infinitesimal gen-

erator as simply the generator.

The domain D(A0) of A0 is the set of all x ∈ X such that the limit in

(2.2) above exists. Then D(A0) is a linear subspace and A0 is a linear operator.

In general, the operator A0 may not be closed, nor densely defined. But, D(A0)

is always non-empty:

Lemma 2.2.1. D(A0) is non-empty.

Proof. For y ∈ X and 0 < α < β <∞, set xα,β =

∫ β

α

T (t)y dt, which exists by

Lemma 1.2.5. We shall establish that limη→0 Aηxα,β exists, thus proving the

result. Indeed for η > 0, by a change of variable in one of the integrals, we have

2.2. The generator 23

that

Aηxα,β =1

η

∫ β

α

T (t+ η)y dt− 1

η

∫ β

α

T (t)y dt

=1

η

∫ β+η

β

T (t)y dt− 1

η

∫ α+η

α

T (t)y dt

=1

η

∫ η

0

T (t+ β)y dt− 1

η

∫ η

0

T (t+ α)y dt

−→ (T (β)y − T (α)y),

as η → 0+, by Lemma 2.1.2. �

Next, we set, for α > 0,

Xα = T (α)(X) and X0 =⋃α>0

Xα.

The semigroup property clearly implies that Xβ ⊂ Xα, for α < β, and

X0 is the smallest linear subspace containing the range spaces of T.

Theorem 2.2.2. If T is a semigroup which is strongly continuous for t > 0, then

for all x ∈ D(A0), we have T (t)x ∈ D(A0) and

T (t)A0x = A0T (t)x =d

dtT (t)x. (2.3)

Proof. For x ∈ D(A0) and t, η > 0,

T (t+ η)x− T (t)x

η= T (t)Aηx = AηT (t)x.

Since limη→0+ Aηx exists and equals A0x and T (t) is bounded,

limη→0+ T (t)Aηx exists and equals T (t)A0x. This implies that both the

limits

limη→0+

AηT (t)x and limη→0+

T (t+ η)x− T (t)x

η

exist. It follows therefore that T (t)x ∈ D(A0) and

limη→0+

T (t+ η)x − T (t)x

η= T (t)A0x = A0T (t)x.

Thus the right hand derivative of T (·)x exists at t and equals A0T (t)x =

T (t)A0x. Also, for η > 0 sufficiently small and t > 0,

T (t− η)x− T (t)x

−η = T (t− η)Aηx. (2.4)

24 C0-semigroups

The term on the right in equation (2.4) approaches T (t)A0x as η → 0+, due to

the strong continuity of T at t and Lemma 1.5.3. Thus, the left hand derivative

of T (t)x also exists for t > 0 and we have

d

dtT (t)x = T (t)A0x.

�

Theorem 2.2.3. If T = {T (t)}t>0 is a strongly continuous semigroup, then

(a) D(A0) is dense in X0;

(b) D(A0) = X0 and

(c) the range of A0 is contained in X0.

Proof. (a) Let x ∈ X0 =⋃

α>0 T (α)X. Then there exists y ∈ X and α > 0

such that x = T (α)y. For β > α > 0, xα,β defined in the proof of Lemma

2.2.1 exists and is in D(A0). Moreover,

xα,β = T (α/2)

∫ β

α

T (t− α/2)y dt = T (α/2)

∫ β−α/2

α/2

T (t)y dt

= T (α/2)xα/2,β−α/2 ∈ X0.

Since limβ→α

1

β − αxα,β = T (α)y = x, it follows that D(A0) is dense in X0,

that is,

X0 ⊂ D(A0) ∩X0. (2.5)

(b) Let x ∈ D(A0). Then by Theorem 2.2.2,

T (t)x− x =

∫ t

0

T (s)A0x dt

(see Lemma 1.2.5) and therefore, limt→0+ T (t)x exists and equals x. Thus

x ∈ X0 and D(A0) ⊂ X0. This along with (2.5) implies that D(A0) = X0.

(c) If x ∈ D(A0) ⊂ X0, then Aηx ∈ X0 so that A0x ∈ X0.

�

Since by definition of X0, T (t) maps X0 into X0, the restriction T (t)∣∣X0

is

itself a strongly continuous semigroup for t > 0. Also, if we assume additionally,

that T (0) = I, then X0 = X, and then there is continuity for t ≥ 0. However,

in general the following holds.


Theorem 2.2.4. Let {T (t)}t>0 ∈ B(X) be a semigroup. Assume that t �→ T (t)

is strongly measurable on (0,∞) and limt→0+ T (t)x = Jx exists for each x ∈ X.

Then J is an idempotent, that is, J2 = J, RanJ = X0 and JT (t) = T (t)J =

T (t) for t > 0. In fact, Jx = x for all x ∈ X0.

Proof. By Theorem 1.5.4, T (t) is strongly continuous on (0,∞). By the Uniform

Boundedness Theorem, J is a bounded operator on X. Moreover, for t, s > 0

and x ∈ X,T (s)T (t)x = T (s+t)x = T (t)T (s)x. Letting t→ 0+ in this equation

yields,

T (s)Jx = T (s)x = JT (s)x, for all s > 0, x ∈ X. (2.6)

Again, letting s → 0+ in (2.6) above yields J2x = Jx for all x ∈ X. Now

for x ∈ X,T (t)x ∈ X0 for all t > 0. Therefore, Jx ∈ X0. Thus, J(X) ⊂ X0.

Now let x ∈ D(A0). Then T (t)x → x as t → 0+. This implies that Jx = x

for all x ∈ D(A0). Since J is bounded in X we have that Jx = x for all

x ∈ D(A0) = X0. Therefore, X0 ⊂ J(X0) ⊂ J(X). Hence, J(X) = X0. �

In view of the above theorem, there are the following alternatives: if

T (t)x → x as t → 0+ for all x ∈ X, then J = I = T (0) and X0 = X, or

{T (t)}t≥0 is a strongly continuous semigroup for t ≥ 0, and J �= I. In the

second case, we can restrict T (t) to X0, and work with this space.

From now on, we shall work with strongly continuous semigroups

{T (t)}t≥0 with T (0) = I. As mentioned earlier, these are called C0-semigroups.

We formalise the definitions in the following.

Definition 2.2.5. A family {T (t)}t≥0 ∈ B(X) is a C0-semigroup if

1. T (t+ s) = T (t)T (s) = T (s)T (t) for all s, t ≥ 0.

2. t �→ T (t) is strongly continuous on [0,∞), that is, limt→s T (t)x = T (s)x,

for all s ≥ 0 and for each x ∈ X.

3. T (0) = I.

As has been observed at the end of Chapter 1, property (2) in Definition

2.2.5 is equivalent to strong continuity at t = 0, that is, for each x ∈ X,

limt→0 T (t)x = T (0)x.

26 C0-semigroups

Definition 2.2.6. The (infinitesimal) generator of a C0-semigroup {T (t)}t≥0 on

a Banach space X is defined as follows :

D(A) = {x ∈ X : lim

t→0+t−1(T (t)x− x) exists

}Ax = lim

t→0+t−1(T (t)x− x)

Note from A.1.4 that for a closed operator A, the resolvent set of A is

given by ρ(A) = {λ ∈ C : (λ−A)−1 ∈ B(X)} and R(λ,A) = (λ−A)−1 is called

the resolvent of A, while the spectrum of A is given by σ(A) := C \ ρ(A).The next theorem sums up some of the important properties of a C0-

semigroup and its generator.

Theorem 2.2.7. Let T = {T (t)}t≥0 be a C0-semigroup defined on the Banach

space X . Then the following properties hold.

(a) There exists M > 0 and β ∈ R such that ‖T (t)‖ ≤ Meβt for all t ≥ 0,

that is, T (t) is exponentially bounded.

(b) T (t)Ax = AT (t)x =d

dtT (t)x for all x ∈ D(A), where A is the generator

of T.

(c)

∫ t

0

T (s)x ds ∈ D(A), for all t ≥ 0, x ∈ X. Furthermore,

T (t)x− x =

⎧⎪⎪⎨⎪⎪⎩A

∫ t

0

T (s)x ds if x ∈ X∫ t

0

T (s)Axds if x ∈ D(A).(2.7)

(d) The generator A of the semigroup is a densely defined, closed linear op-

erator.

(e) The half plane Hβ = {z ∈ C : Re z > β} is contained in the resolvent set

ρ(A) and the resolvent R(z, A) is given by

R(z, A)x =

∫ ∞

0

e−ztT (t)x dt, for all z ∈ Hβ .

(f) ‖R(z, A)n‖ ≤M(Re z − β)−n for Re z > β and n = 1, 2 . . . .

Proof. (a) The semigroup property of T combined with the hypothesis of

strong continuity at 0 implies (a) on using Theorem 1.5.9, while


(b) follows from Theorem 2.2.2.

(c) Let x ∈ X. Writing x0,t =

∫ t

0

T (s)x ds as before, and following the proof

of Lemma 2.2.1, we have that

1

h

(T (h)x0,t − x0,t

)=

1

h

∫ t

0

T (s+ h)x ds−∫ t

0

T (s)x ds

=1

h

∫ t+h

t

T (s)x ds− 1

h

∫ h

0

T (s)x ds

−→ (T (t)x− x), as h→ 0+, (2.8)

by Lemma 2.1.2. Therefore, x0,t ∈ D(A), and

T (t)x− x = A

∫ t

0

T (s)x ds = Ax0,t.

Now suppose x ∈ D(A). Then by Theorem 2.2.2, T (t)x ∈ D(A), andwe set,

g(t) = AT (t)x = T (t)Ax and

gh(t) =1

h(T (t+ h)x− T (t)x).

Then the strong continuity of T implies that g, gh are continuous on [0,∞)

and using (a), we have, for t ≥ 0, that

‖gh(t)− g(t)‖ = ∥∥T (t)( 1h

(T (h)x− x

)−Ax)∥∥

≤Meβt∥∥ 1h

(T (h)x− x

)−Ax∥∥.

Thus gh → g as h → 0, uniformly on compact subintervals of [0,∞) and

by (2.8) one gets that

T (t)x−x = limh↓0

∫ t

0

gh(s) ds =

∫ t

0

g(s) ds =

∫ t

0

AT (s)x ds =

∫ t

0

T (s)Axds.

(d) In view of Theorem 2.2.3 and the discussion following immediately after

Theorem 2.2.4, D(A) = X0 = X, so that A is densely defined. To see that

A is closed, let {xn}n be a sequence in D(A), converging to x and suppose

that Axn → y as n→∞ for some y ∈ X. By (c) we have that

T (t)xn − xn =

∫ t

0

T (s)Axn ds (2.9)

28 C0-semigroups

for all n = 1, 2, 3... Since T (t) is bounded, T (t)xn → T (t)x as n → ∞ so

that the left hand side of (2.9) converges to T (t)x− x as n→∞. On the

other hand,

∥∥ ∫ t

0

(T (s)Axn − T (s)y) ds∥∥ ≤M‖Axn − y‖

∫ t

0

eβs ds→ 0 as n→∞.

Therefore, T (t)x− x =

∫ t

0

T (s) y ds, so that by Lemma 2.1.2

Aηx = η−1

∫ η

0

T (s)y ds→ y as η → 0+.

This implies that x ∈ D(A) and Ax = y. Thus A is closed.

(e) Let z ∈ Hβ and x ∈ D(A). Thend

dtT (t)x = T (t)Ax = T (t)(A− z)x+ zT (t)x.

For x∗ ∈ X∗, we have thatd

dte−zt〈x∗, T (t)x〉 = e−zt〈x∗, T (t)(A − z)x〉.

Since T (0) = I, we have on integrating, that for x ∈ D(A),

e−zt〈x∗, T (t)x〉 − 〈x∗, x〉 =∫ t

0

e−zs〈x∗, T (s)(A− z)x〉 ds

or, 〈x∗, x〉 = e−zt〈x∗, T (t)x〉 −∫ t

0

e−zs〈x∗, T (s)(A− z)x〉 ds.

Thus, for all x∗ ∈ X∗, x ∈ D(A),

|〈x∗, x〉| ≤M‖x∗‖{e(β−Re z)t‖x‖+∫ t

0

e(β−Re z)s‖(A− z)x‖}.Therefore, by an application of the Hahn-Banach theorem, one has that

‖x‖ ≤M{e(β−Re z)t‖x‖+ ‖(A− z)x‖(Re z − β)−1[1− e−(Re z−β)t]

}.

Letting t → ∞ in the above equation, we get, since Re z > β, that for

x ∈ D(A),‖x‖ ≤M‖(A− z)x‖(Re z − β)−1

so that,

‖(A− z)x‖ ≥M−1(Re z − β)‖x‖, (2.10)


for all z ∈ C such that Re z > β. Thus, (A−z) is injective. Next we show

that Ran (A− z) is a closed subspace of X for Re z > β. Let {xn}n∈N be

a sequence in Ran (A − z) converging to x ∈ X as n → ∞. Then there

exist yn ∈ D(A) such that xn = (A− z)yn, n = 1, 2... and (2.10) leads to

‖xn − xm‖ ≥M−1(Re z − β)‖yn − ym‖.

Therefore, {yn} is Cauchy and hence converges to some y ∈ X, while

{(A− z)yn} converges, by assumption to x. Since A is closed, this implies

that y ∈ D(A) and (A − z)y = x. Thus, x lies in Ran (A − z), so that

Ran (A− z) is closed.

Now suppose that x∗ ∈ X∗ is such that 〈x∗, (A − z)y〉 = 0, for all

y ∈ D(A). For such an x∗, and any y ∈ D(A),

d

dt〈x∗, T (t)y〉 = 〈x∗, AT (t)y〉

= 〈x∗, (A− z)T (t)y〉+ z〈x∗, T (t)y〉 = z〈x∗, T (t)y〉,

since by (b), T (t) maps D(A) into itself. This implies that

〈x∗, T (t)y〉 = ezt〈x∗, y〉.

Therefore,

∣∣〈x∗, y〉∣∣ ≤M‖x∗‖‖y‖e−(Re z−β)t → 0 as t→∞, for Re z > β.

Thus 〈x∗, y〉 = 0, for all y ∈ D(A). Since A is densely defined, that is,

D(A) = X, it follows that 〈x∗, x〉 = 0 for all x ∈ X. This forces x∗ to be

0.

If Ran (A− z) �= X, then there exists x0 �= 0 such that x0 /∈Ran (A− z). The Hahn-Banach theorem then shows that there exists

x∗ ∈ X∗ such that 〈x∗, x0〉 = 1, 〈x∗, x〉 = 0 for all x ∈ Ran (A− z). But

this implies, from the discussion in the last paragraph that x∗ = 0, lead-

ing to a contradiction. Therefore, Ran (A− z) = X and the closedness

of Ran(A − z) implies that Ran (A − z) = X. Thus Hβ ⊂ ρ(A) and

‖(A− z)−1‖ ≤M(Re z − β)−1.

30 C0-semigroups

Suppose again that Re z > β and x ∈ D(A), and let a > 0. Since A

is closed, it follows from Lemma 1.2.5 that

∫ a

0

e−ztT (t)x dt ∈ D(A) and

A

∫ a

0

e−ztT (t)x dt =

∫ a

0

e−ztAT (t)x dt =

∫ a

0

e−zt d

dtT (t)x dt

= e−zaT (a)x− x+ z

∫ a

0

e−ztT (t)x dt.

Letting a → ∞ in the above shows that A

∫ ∞

0

e−ztT (t)x dt exists and

that

A

∫ ∞

0

e−ztT (t)x dt = −x+ z

∫ ∞

0

e−ztT (t)x dt

or, (z −A)

∫ ∞

0

e−ztT (t)x dt = x.

where we have used the facts that A is a closed operator and that T is

of type β so that ‖e−ztT (t)x‖ ≤ Me−(Re z−β)t‖x‖. Since z ∈ ρ(A), this

implies that for x ∈ D(A),

(z −A)−1x =

∫ ∞

0

e−ztT (t)x dt. (2.11)

Since (z−A)−1 is bounded and since the integral in (2.11) is well defined

for all x ∈ X it follows that (2.11) holds for all x ∈ X.

(f) The function z �→ R(z, A) ∈ B(X) is strongly differentiable (in fact, in

operator norm) for Re z > β. This can be seen from (2.11) by differenti-

ating inside the integral in the right hand side, which is permitted by an

application of the Dominated Convergence Theorem using the property

(a). Then one has, for x ∈ D(A), that

d

dz

[R(z, A)x

]= −(z −A)−2x = −

∫ ∞

0

te−ztT (t)x dt. (2.12)

Differentiating (2.11) (n−1) times with respect to z in the half-plane Hβ ,

gives

(n− 1)!(z −A)−nx =

∫ ∞

0

e−zttn−1T (t)x dt,


where the interchange of differentiation in z and integration with respect

to t is done as before. Therefore, for all Re z > β,

‖(z −A)−n‖ ≤ M

(n− 1)!

∫ ∞

0

e−(Re z−β)ttn−1dt

≤ M

(n− 1)!(Re z − β)nΓ(n) =

M

(Re z − β)n.

�

Remark 2.2.8. It is not difficult to see that the generator A of a C0 semigroup

{T (t)}t≥0 defined on a Banach space X is a bounded operator on X if and

only if the semigroup is norm continuous on [0,∞), (that is, the map t→ T (t)

from [0,∞) to B(X) is continuous with respect to the operator norm topology

in B(X)). In such a case, the semigroup {T (t)}t≥0 is referred to as a norm

continuous or uniformly continuous semigroup.

So far, we have talked about operator valued functions T defined on [0,∞)

satisfying the semigroup property. It is perfectly possible to talk of operator

valued functions having all of R as their domain and demanding that they

satisfy properties similar to those in Definition 2.2.5. This would, of course,

result in a much more restricted class of operators; such a family of operators

is called a C0 group. Precisely, we have

Definition 2.2.9. A C0-group on a Banach space X is a family {T (t)}t∈R of

operators in B(X) satisfying the following properties.

1. T (t+ s) = T (t)T (s) = T (s)T (t) for all s, t ∈ R.

2. t �→ T (t) is strongly continuous on R, that is, limt→s T (t)x = T (s)x, for

all s ∈ R and every x ∈ X.

3. T (0) = I.

The generator A of this group is defined by setting

D(A) = {x ∈ X : lim

t→0t−1(T (t)x− x) exists

},

Ax = limt→0

t−1(T (t)x− x).

Remark 2.2.10. Note that the limit in the definition of the generator of a

C0-group is a two-sided one, not just right-sided, which is the case for C0-

semigroups. Also see Remark 2.3.4(2).

32 C0-semigroups

2.3 The Hille-Yosida Theorem

Theorem 2.2.7 shows that for a linear operator A to be the generator of a C0-

semigroup it is necessary that A be densely defined and closed. But this is not

sufficient. Conditions like (e) and (f) of Theorem 2.2.7 are not only necessary

but also sufficient, as is shown by the Hille-Yosida Theorem. This is perhaps

the most important result in the theory of operator semigroups and was proven

independently by Hille [12] and Yosida [26].

Theorem 2.3.1 (Hille-Yosida). Let A be a densely defined, closed linear oper-

ator on X. Let Hβ = {z ∈ C : Re z > β} ⊂ ρ(A) for some β ∈ R and suppose

that there exists M > 0 such that∥∥(z −A)−n∥∥ ≤M(Re z − β)−n for all z ∈ Hβ and n = 1, 2, ... (2.13)

Then there exists a unique C0-semigroup {T (t)}t≥0 such that A is its generator

and ‖T (t)‖ ≤Meβt.

Proof. Fix β ∈ R, and set B = A − β and w = z − β. Then the hypothesis

imply that H0 ⊂ ρ(B), w −B = z −A and

‖(w −B)−n‖ ≤M(Re w)−n for all w ∈ H0 and n = 1, 2....

So we may assume without loss of generality that β = 0. Set, for n = 1, 2, ...

An = nA(n−A)−1. (2.14)

Then, An ∈ B(X) for each n and for every x ∈ D(A), limn→∞ Anx = Ax.

Indeed, for x ∈ D(A),

‖Anx− Ax‖ = ‖(n(n−A)−1 − 1)Ax‖ = ‖(In − I)Ax‖,

where Inx = n(n−A)−1x. But, for any y ∈ D(A),

‖Iny − y‖ = ‖(n−A)−1Ay‖ ≤ M

n‖Ay‖ → 0 as n→∞.

Since D(A) is dense in X and ‖In‖ ≤M by the hypothesis, it follows that In →I strongly onX as n→∞. Hence, for each x ∈ D(A), Anx→ Ax as n→∞. Let

Tn(t) = etAn be defined by the convergent power series

∞∑k=0

tk

k!(An)

k ∈ B(X).

We note that Tn is an entire function of t for every n ∈ N and

d

dtTn(t) = AnTn(t) and Tn(0) = I.

2.3. The Hille-Yosida Theorem 33

Since An = n(In − I), n ∈ N,

Tn(t) = etn(In−I) = e−ntetnIn

for each n. By (2.13), we have that

∥∥(nIn)m∥∥ =∥∥n2m(n−A)−m

∥∥ ≤Mn2mn−m = Mnm,

for all m ∈ N. Therefore, for t ≥ 0,

∥∥etnIn∥∥ ≤ ∞∑m=0

1

m!tm∥∥(nIn)m∥∥ ≤ ∞∑

m=0

1

m!tmMnm = Ment,

so that ‖Tn(t)‖ ≤ M. Now, Tm(t) = etAm and An, being functions of Im and

In respectively, commute. Therefore, for x ∈ D(A) and n,m ∈ N, one gets that

Tn(t)x− Tm(t)x = Tn(t− s)Tm(s)x∣∣0t= −

∫ t

0

d

ds

[Tn(t− s)Tm(s)x

]ds

=

∫ t

0

Tn(t− s)(An −Am)Tm(s)x ds

=

∫ t

0

Tn(t− s)Tm(s)(Anx−Amx) ds.

Therefore,

‖Tn(t)x − Tm(t)x‖ ≤M2t∥∥Anx−Amx

∥∥→ 0 as n,m→∞.

Thus {Tn(t)x}n is a Cauchy sequence for each t ∈ [0,∞) and x ∈ D(A).Moreover, the sequence is uniformly Cauchy for all t in compact subsets of

[0,∞). Since ‖Tn(t)‖ is uniformly bounded and D(A) is dense in X it follows

that {Tn(t)x}n is a Cauchy sequence for each t ∈ [0,∞) and x ∈ X. Set

T (t)x = limn→∞Tn(t)x,

for all x ∈ X and t ∈ [0,∞).As we have noted earlier, the convergence Tn(t)x→T (t)x as n → ∞ is uniform for all t in compact subsets of [0,∞), for each

x ∈ D(A). This uniform convergence extends to all of x ∈ X. Indeed, for

ε > 0, x ∈ X, and a compact subset K of [0,∞), one can choose a y ∈ D(A),such that ‖x− y‖ < ε

4Mand an n0 depending on ε and y such that

‖Tn(t)y − T (t)y‖ < ε

2for all n > n0

34 C0-semigroups

and t ∈ K. Then, for all t ∈ K,

‖Tn(t)x− T (t)x‖ ≤ ‖(Tn(t)− T (t))(x− y)‖+ ‖(Tn(t)− T (t))y‖≤ 2M‖x− y‖+ ‖(Tn(t)− T (t))y‖ < ε

for all n > n0 and for all t ∈ K, leading to the strong convergence, uniformly

with respect to t, in compact subsets of [0,∞). Since Tn(t)Tn(s)x = Tn(t+ s)x

for all n ∈ N, x ∈ X, taking strong limit as n→∞ in the above, it follows that

T (t)T (s) = T (t+ s) for t, s ≥ 0. Furthermore, for any x ∈ X,

limt→0+

T (t)x = limt→0+

limn→∞ Tn(t)x = lim

n→∞ limt→0+

Tn(t)x = x

where the interchange in the order of limits is permissible due to uniform con-

vergence, and T (0)x = limn→∞ Tn(0)x = x. Thus, {T (t)}t≥0 is a C0-semigroup.

We show next that A is the generator of this semigroup. For x ∈ D(A) and

n ∈ N we have from Theorem 2.2.7 (c) that

Tn(t)x − x =

∫ t

0

Tn(s)Anx ds. (2.15)

The left hand side of (2.15) converges to T (t)x− x. On the right hand side of

(2.15), Anx→ Ax while Tn(s)→ T (s), strongly and uniformly for s ∈ [0, t], as

n→∞. Therefore, taking limit as n→∞ in (2.15), we obtain,

T (t)x− x =

∫ t

0

T (s)Axds for all x ∈ D(A),

and hence, by Lemma 2.1.2,

limt→0+

t−1(T (t)x− x) = limt→0+

t−1( ∫ t

0

T (s)Axds)= Ax for all x ∈ D(A).

Thus Ax = Ax for all x ∈ D(A), where A denotes the generator of the semi-

group T (t) constructed above. Now, let x ∈ D(A) and set v = (A − 1)x. By

hypothesis, 1 ∈ ρ(A). Therefore, there exists w ∈ D(A) such that v = (A−1)w.

This implies that,

(A− 1)x = (A− 1)w = (A− 1)w. (2.16)

By Theorem 2.2.7 (e), and since ‖T (t)‖ ≤ M it follows that 1 ∈ ρ(A), that is,

(A− 1)−1 ∈ B(X). Therefore, (2.16) implies that x = w ∈ D(A). Thus A = A.

2.3. The Hille-Yosida Theorem 35

Finally, for uniqueness, let {T (t)}t≥0 be another semigroup generated by

A. For 0 < s < t, and x ∈ D(A),d

ds(T (t− s)T (s)x) = −T (t− s)AT (s)x+ T (t− s)AT (s)x = 0.

Thus, T (t − s)T (s)x is independent of s for 0 < s < t. Therefore, for

x ∈ D(A),

T (t)x = lims→0+

T (t− s)T (s)x = lims→t−

T (t− s)T (s)x = T (t)x.

The boundedness of T and T allows us to extend this equality to all x ∈ X.

�

The following version of the Hille-Yosida Theorem is often useful and its

proof is left as an exercise (Exercise 2.3.2).

Theorem 2.3.3. A linear operator A is the infinitesimal generator of a

(i) contraction C0-semigroup {T (t)}t≥0 if and only if

(a) A is closed and densely defined in X ;

(b) the resolvent set ρ(A) of A contains the interval (0,∞) and

∥∥R(λ,A)∥∥ ≤ λ−1 for all λ > 0.

(ii) C0-semigroup {T (t)}t≥0 satisfying ‖T (t)‖ ≤Meβt if and only if

(a) A is closed and densely defined in X ;

(b) the resolvent set ρ(A) of A contains the interval (β,∞) and

∥∥R(λ,A)n∥∥ ≤M(λ− β)−n for all λ > β, and n = 1, 2, . . . .

Remark 2.3.4. 1. In Theorem 2.3.1, instead of approximating T (t) by etAn ,

where An = nA(n−A)−1, we can try a different approximation:

Vn(t) =

(1− tA

n

)−n

.

This is not a semigroup for any n. But, by the hypotheses of Theorem

2.3.1,

(i) ‖Vn(t)‖ ≤M, ∀n and t ≥ 0;

36 C0-semigroups

(ii) Vn(t) is differentiable with respect to t for all t > 0, with

V′

n(t) = A

(1− tA

n

)−n−1

∈ B(X);

(iii) Vn(t) is not, in general, differentiable in B(X) at t = 0, but is strongly

continuous:

Vn(t)→ Vn(0) = I as t→ 0+;

(iv) {Vn(t)x}n is strongly Cauchy for each t ≥ 0 and x ∈ X, uniformly

for t in compact subsets of [0,∞).

The verification of the above claims and using them, the writing of

an alternative proof for the Hille-Yosida Theorem are left as an exercise

(Exercise 2.3.5).

2. The approximation in (1) above may be used to show that A is the gen-

erator of a C0-group {T (t)}t∈R if and only if ±A each generate respec-

tively the C0-semigroup {T±(t)}t≥0 where T+(t) := T (t), for all t ≥ 0,

and T−(t) := T (−t), for all t ≥ 0. The details of the proof are left as an

exercise (Exercise 2.3.6).

2.4 Adjoint semigroups

A C0-semigroup {T (t)}t≥0 defined on a Banach space X induces in a natural

way a family of operators {T ∗(t)}t≥0 on the dual space X∗, where T ∗(t) is the

adjoint of T (t) for each t ≥ 0. This family clearly satisfies the semigroup law:

〈T ∗(t)T ∗(s)x∗, x〉 = 〈T ∗(s)x∗, T (t)x〉 = 〈x∗, T (s)T (t)x〉= 〈x∗, T (t+ s)x〉 = 〈T ∗(s+ t)x∗, x〉

for all t, s ≥ 0, x∗ ∈ X∗ and x ∈ X. Also, 〈T ∗(0)x∗, x〉 = 〈x∗, T (0)x〉 = 〈x∗, x〉so that T ∗(0) = I. This semigroup may not be strongly continuous in general.

However, if X is reflexive, then {T ∗(t)}t≥0 is a C0-semigroup with generator

A∗.

Theorem 2.4.1. Let X be a reflexive Banach space. If {T (t)}t≥0 is a C0-

semigroup on X, with generator A, then {T ∗(t)}t≥0 is a C0-semigroup with

generator A∗. Conversely, if A is a closed, densely defined operator on X and

2.4. Adjoint semigroups 37

β ∈ R is such that A satisfies the resolvent estimates as in (2.13) for z ∈Hβ = {z ∈ C : Re z > β} ⊂ ρ(A), then so does A∗. Also, if {T (t)}t≥0 is the

semigroup generated by A, then {T ∗(t)}t≥0 is the semigroup generated by A∗.

Proof. Since X is reflexive, we shall identify x ∈ X with its embedded image

in X∗∗ without any new notation for the same. We have already seen that

T ∗(t)T ∗(s) = T ∗(t + s) and T ∗(0) = I. Since {T (t)}t≥0 is a C0-semigroup, it

is exponentially bounded. Suppose that ‖T (t)‖ ≤ Meβt for some β ∈ R and

M > 0. Then, for x ∈ X and x∗ ∈ X∗,∣∣〈T ∗(t)x∗, x〉∣∣ = ∣∣〈x∗, T (t)x〉∣∣ ≤ ‖x∗‖‖x‖Meβt.

This implies that∥∥T ∗(t)

∥∥ ≤Meβt. Also, for t ≥ s ≥ 0,

⟨T ∗(t)x∗ − T ∗(s)x∗, x

⟩=⟨x∗, (T (t)− T (s))x

⟩→ 0

as t → s. Therefore by the reflexivity of X, {T ∗(t)}t≥0 is a weakly continu-

ous semigroup. It follows, on using Corollary 1.5.6, that {T ∗(t)} is strongly

continuous. Thus, {T ∗(t)}t≥0 is a C0-semigroup and⟨(T ∗(t)− I)

tx∗, x

⟩=⟨x∗,

(T (t)− I)

tx⟩→ ⟨

x∗, Ax⟩

as t → 0+ for all x ∈ D(A). Let A be the generator of {T ∗(t)}t≥0. Then, for

x∗ ∈ D(A), ⟨Ax∗, x

⟩=⟨x∗, Ax

⟩,

which implies A ⊂ A∗ since D(A) is dense. For all x∗ ∈ D(A∗) and x ∈ D(A),⟨T ∗(t)x∗ − x∗, x

⟩=⟨x∗, T (t)x− x

⟩=

∫ t

0

⟨x∗, AT (s)x

⟩ds

=

∫ t

0

⟨T ∗(s)A∗x∗, x

⟩ds,

leading to the identity T ∗(t)x∗ − x∗ =

∫ t

0

T ∗(s)A∗x∗ds, since D(A) is dense

in X. But this implies, by Lemma 2.1.2, that x∗ ∈ D(A) and Ax∗ = A∗x∗.

Therefore, A∗ ⊂ A and hence A∗ = A.

�

The following is an example of a C0-semigroup whose adjoint is not a

C0-semigroup.

38 C0-semigroups

Example 2.4.2. Let X be the Banach space BUC(R+) of bounded, uniformly

continuous functions on [0,∞) equipped with the supremum norm. Define the

family {T (t)}t≥0 on X as follows:

(T (t)f)(s) = f(s+ t) for all f ∈ X and s, t ≥ 0.

We have seen in Example 1.5.10 that {T (t)}t≥0 is a C0-semigroup. Let A denote

its generator. Set

BUC1(R+) = {f ∈ BUC(R+), f is differentiable, f ′ ∈ BUC(R+)}.

We will show that D(A) = BUC1(R+), Af = f ′ and that T ∗ is not a C0-

semigroup.

Now f ∈ D(A), implies that limt→0+T (t)f − f

texists in BUC(R+), that

is, there exists a g ∈ BUC(R+) such that limt→0+T (t)f − f

t= g in the supre-

mum norm. It follows therefore that for each s ≥ 0,

(T (t)f)(s)− f(s)

t− g(s) =

f(t+ s)− f(s)

t− g(s)→ 0

as t → 0. Thus we may conclude that f is differentiable on (0,∞) and for

all s > 0, f ′(s) = g(s) = Af(s). In other words, f ∈ BUC1[0,∞). Therefore,

D(A) ⊂ BUC1[0,∞). On the other hand, if f ∈ BUC1[0,∞), then

f(t+ s)− f(s) =

∫ t

0

f ′(u+ s)d u.

Using this, we obtain that∣∣∣f(t+ s)− f(s)

t− f ′(s)

∣∣∣ ≤ 1

t

∫ t

0

‖(T (u)− I)f ′‖du→ 0 as t→ 0

due to the strong continuity of the semigroup and Lemma 2.1.2. Thus, f ∈ D(A)and f ′ = Af.

Let μ = δa, the delta measure concentrated at a ∈ R. It is easy to see

that μ is in the dual space of BUC(R+). Then, for f ∈ X,

〈T ∗(t)μ, f〉 = 〈μ, T (t)f〉 = (T (t)f)(a) = f(a+ t) = 〈δa+t, f〉.

Therefore, T ∗(t)μ = δa+t, so that∥∥(T ∗(t)− I)μ∥∥ =

∥∥δa+t − δa∥∥ = 2

for all t > 0. Thus T ∗ is not strongly continuous, and it follows that BUC(R+)

is not a reflexive Banach space.

2.5. Examples of C0-semigroups and their generators 39

2.5 Examples of C0-semigroups and their generators

Example 2.5.1. Let X = Lp(R+) for some p be such that 1 ≤ p <∞ and set

(T (t)f)(s) = f(s+ t) for all f ∈ X and s, t ≥ 0.

Then {T (t)}t≥0 is a C0-semigroup with generator A given by

D(A) = {f ∈ Lp(R+) : f absolutely continuous and f ′ ∈ Lp(R+)}= W 1,p(R+) (see Appendix A.3),

Af = f ′. (2.17)

For f ∈ Lp(R+),

‖(T (t)− I)f‖pp =

∫ ∞

0

|f(s+ t)− f(s)|p ds.

Recall that C∞c (R+), the linear space of arbitrarily often differentiable (or

smooth) functions on R+ with compact support, is dense in Lp(R+) ([18,

Proposition 5.5.9 ]). Let f ∈ C∞c (R+) and suppose that the support of

f, supp f ⊂ [a, b] for some 0 < a < b <∞. Then, for |t| < 1,

∫ ∞

0

|f(s+ t)− f(s)|p ds =

∫ b+1

a−1

|f(s+ t)− f(s)|p ds.

Since f(s+ t)→ f(s) as t→ 0+ pointwise, and f is bounded, it follows by the

Dominated Convergence Theorem that

limt→0+

‖(T (t)− I)f‖pp = limt→0+

∫ ∞

0

|f(s+ t)− f(s)|p ds = 0.

This is true for all f ∈ C∞c (R+). The density of C∞

c (R+) now allows us to

extend the above convergence to all f ∈ Lp(R+), on observing that ‖T (t)‖ ≤ 1

for all t ≥ 0. Thus T is a strongly continuous contraction semigroup on Lp(R+).

We shall now determine its generator A. Recall that (see Appendix A.3)

W 1,p(R+) ={f ∈ Lp[0,∞) :

∫ ∞

0

|f ′(s)|p ds <∞},

and that the two sets on the right hand side of (2.17) coincide due to Lemma

A.3.2.

40 C0-semigroups

Let φ ∈ C∞c (R+) with suppφ ⊂ [c, d] for 0 < c < d < ∞, and let

f ∈ D(A). Then

〈t−1(T (t)f − f), φ〉 =∫ ∞

0

t−1[f(s+ t)− f(s)]φ(s) ds

=

∫ ∞

0

f(s)[φ(s− t)− φ(s)

t+ φ

′(s)

]ds

−∫ ∞

0

f(s)φ′(s) ds−

∫ t

0

f(s)φ(s− t)

tds (2.18)

= I1 −∫ ∞

0

f(s)φ′(s) ds+ I2, (2.19)

where I1 and I2 represent the first and the last integrals respectively appearing

on the right hand side of (2.18). Note that φ(s− t) = 0 for 0 ≤ s ≤ t because of

the support properties of φ, rendering I2 = 0. Since both the sets suppφ and

suppφ′ ⊂ [c, d] and since t−1(φ(s− t)−φ(s)) converges to −φ′(s) uniformly in

s ∈ [c, d], as t → 0+, I1 converges to 0 as t → 0+. Hence, taking the limit as

t→ 0+ in (2.19) we obtain that

〈Af, φ〉 = −∫ ∞

0

f(s)φ′(s) ds.

Next, we set g(s) =

∫ s

0

(Af)(τ) dτ which makes g a well defined absolutely

continuous function and an integration by parts leads to the equality

−∫ ∞

0

f(s)φ′(s) ds = 〈Af, φ〉 = −

∫ ∞

0

g(s)φ′(s) ds

for all φ ∈ C∞c (R+). This implies that g = f or equivalently, Af = f ′. There-

fore, D(A) ⊂W 1,p(R+).

Conversely, if f ∈ W 1,p(R+), then f ′ exists in Lp(R+), and we have from

(2.19) that for φ ∈ C∞c (R+),

〈f ′, φ〉 = −∫ ∞

0

f(s)φ′(s) ds = −I1 + 〈t−1(T (t)f − f), φ〉 − I2.

Since I1, I2 → 0 as t→ 0+, it follows that

limt→0+

〈t−1(T (t)f − f), φ〉 = −〈f, φ′〉 = 〈f ′, φ〉

for all φ ∈ C∞c (R+). Hence f ∈ D(A), and Af = f ′.


Remark 2.5.2. (1) The semigroups of operators in Example 2.5.1 above and

Example 2.4.2 are called translation semigroups or shifts (left translations

or left shifts to be precise). Different translation semigroups with distinct

generators can be constructed by changing the underlying Banach space.

For example, one could replace Lp(R+) or BUC(R+) in the above exam-

ples by Lp(R) or BUC(R) respectively, or with C0(R), C0(R+) and other

such function spaces. Here we shall restrict our attention to Lp and BUC

spaces. In both theses cases, the generator is again differentiation, but

with suitably altered domains. The verification of these properties is left

as an exercise (Exercise 2.5.3).

(a) Let X = BUC(R), and (T (t)f)(s) = f(s + t) for all f ∈ X and

s, t ∈ R. Then {T (t)}t∈R is a C0-group of isometries with generator

A given by Af = f ′, for all f ∈ D(A), where

D(A) = {f ∈ BUC(R), f is differentiable, f ′ ∈ BUC(R)}= BUC1(R).

(b) Let X = Lp(R) where 1 ≤ p < ∞ and let (T (t)f)(s) = f(s+ t) for

all f ∈ X and s, t ∈ R. Then {T (t)}t∈R is a C0-group of isometries

with generator A given by Af = f ′, ∀ f ∈ D(A) = W 1,p(R) (see

Appendix A.3).

(2) Let X = BUC(R) or Lp(R). The inverse of T (t), the left shift operator on

X , is the right shift operator S(t), given by (S(t)f)(s) = f(s− t) for all

f ∈ X and t, s ∈ R. Moreover, {S(t)}t∈R forms a C0-group of isometries

with generator −A, where A is the generator of the left shift group defined

in (1a) or (1b) respectively.

(3) The right translation semigroup on Lp(R+) is defined as follows:

(S(t)f)(s) =

⎧⎨⎩f(s− t) if s− t ≥ 0

0 if s− t < 0.(2.20)

Then {S(t)}t≥0 defines a C0-semigroup of isometries on Lp(R+) with gen-

erator A given by D(A) = {f ∈ W 1,p : f(0) = 0} and (Af)(s) = −f ′(s)

if s > 0, for any f ∈ D(A). Every function in RanS(t) vanishes in the

interval [0, t] and therefore RanS(t) �= Lp(R+). A similar result holds on

BUC(R+).

42 C0-semigroups

(4) Translation semigroups can also be defined on function spaces on finite

intervals using the same ideas as above and making appropriate mod-

ifications. For example, on Lp[a, b], where a, b ∈ R, we define the left

translation semigroup {T (t)}t≥0 by

(T (t)f)(s) :=

⎧⎨⎩f(s+ t) if a ≤ s+ t ≤ b

0 if s+ t > b or s+ t < a.∀ f ∈ Lp[a, b].

(2.21)

Then {T (t)}t≥0 is a C0-semigroup which is nilpotent , that is, T (t) = 0 for

all t ≥ b − a.

Let us now revisit the semigroup in Example 1.5.7. Here {T (t)}t≥0 is a C0-

semigroup and for each t > 0, T (t) is multiplication by the function etq, where

q(s) = −s2, for all s ≥ 0. This is an example of a multiplication semigroup,

which is usually defined on spaces of continuous or measurable functions. In

the next example we look at general multiplication semigroups on Lp spaces.

Recall that for a measurable function q : Ω → C, where (Ω,�, μ) is a σ-finite

measure space, the essential range of q is defined to be

ess ran q :=⋂ε>0

{λ : μ({s ∈ Ω : |q(s)− λ| < ε}) > 0}.

The associated multiplication operator Mq on Lp(Ω, μ) is defined as follows:

D(Mq) : = {f ∈ Lp(Ω, μ) : qf ∈ Lp(Ω, μ)},Mq = qf. (2.22)

Then Mq is a closed and densely defined operator. Mq is bounded if and only

if q ∈ L∞(Ω, μ) and in this case

‖Mq‖ = ‖q‖∞ = sup{|λ| : λ ∈ ess ran q}.

Further, Mq has a bounded inverse if and only if 0 /∈ ess ran q. In such a case,

μ{s ∈ Ω : q(s) = 0} = 0 and then (Mq)−1 = Mψ where ψ(s) = 1/q(s), μ-almost

everywhere. It can be seen that

σ(Mq) = ess ran q, and for λ /∈ σ(Mq), R(λ,Mq) = Mφ, (2.23)

where φ is given by

φ(s) = (q(s)− λ)−1, μ-almost everywhere. (2.24)


We refer the reader to [1] and [21] for a detailed discussion of such operators.

See also Theorem A.1.13.

Example 2.5.4. Let X = Lp(Ω, μ), where (Ω,�, μ) is a σ-finite measure space

and p ∈ [1,∞) is fixed. Further, let q : Ω → C be a measurable function satis-

fying supλ∈ess ran q Reλ <∞. Define T (t) ∈ B(X) by

(T (t)f)(s) = etq(s)f(s), for all s ∈ Ω, t ≥ 0 and f ∈ X.

Then {T (t)}t≥0 is a C0-semigroup with generator A = Mq.

We first check strong continuity of the semigroup {T (t)}t≥0. Let k =

supλ∈ess ran q Reλ. Then we have, for f ∈ X and t > 0, that

‖T (t)f − f‖p =

∫Ω

∣∣(etq(s) − 1)f(s)∣∣p μ(ds),

and since ∣∣etq(s) − 1∣∣ ≤ 1 + etk for 0 ≤ t ≤ 1,

it follows, by using the Dominated Convergence Theorem that ‖T (t)f−f‖ → 0

as t→ 0+. Moreover, it is easy to check that

‖T (t)‖ ≤ etk, ∀ t ≥ 0. (2.25)

Let A be the generator of this semigroup. To show that A = Mq let us first

take f ∈ D(A). For s ∈ Ω, we have

limt→0+

t−1((T (t)f)(s)− f(s)

)= lim

t→0+t−1(etq(s) − 1)f(s) = q(s)f(s). (2.26)

Since f ∈ D(A) implies that limt→0+ t−1((T (t)f)− f

)exists in X = Lp(Ω, μ),

which implies pointwise convergence μ-almost everywhere, possibly for a sub-

sequence, it follows from (2.26) that the limit equals Mqf and Mqf ∈ Lp(Ω, μ).

Thus f ∈ D(Mq) and Af = Mqf. Hence, A ⊂Mq. We now show that Mq ⊂ A.

Let f ∈ D(Mq). Since∣∣∣etq(s) − 1− tq(s)∣∣∣ = ∣∣∣q(s)∫ t

0

(erq(s) − 1

)dr∣∣∣ ≤ |q(s)|(t+ ∫ t

0

erRe q(s) dr)

≤ |q(s)|∣∣t+ tetk∣∣,

we have, for μ-almost all s and for 0 < t < 1,∣∣t−1(etq(s) − 1− tq(s)

)f(s)

∣∣ ≤ (1 + ek)|q(s)f(s)|.

44 C0-semigroups

Moreover,

t−1(etq(s) − 1− tq(s)

)→ 0 for μ-almost all s as t→ 0+.

Therefore, by the Dominated Convergence Theorem,∫Ω

∣∣t−1(etq(s) − 1)f(s)− q(s)f(s)∣∣p μ(ds)→ 0 as t→ 0+.

This implies that limt→0+ t−1(T (t)f − f

)= qf in X. Hence f ∈ D(A) and

Af = qf = Mqf. Thus Mq ⊂ A.

The example we discuss next – Gaussian semigroups – are representa-

tives of a very important class of semigroups, called convolution semigroups.

Gaussian semigroups, also called heat semigroups or diffusion semigroups, by

themselves are an important class of C0-semigroups, occurring very frequently

in applications.

Example 2.5.5 (Convolution Semigroups). Convolution semigroups occur in

many areas of applications, particularly in probability theory and are also of

interest by themselves. Special examples of some of them, particularly the heat

semigroup, play important roles in the theory of Brownian motion, of diffusion

processes and in geometry.

Definition 2.5.6. A convolution semigroup on Rd is a family of probability

measures {μt}t∈R satisfying, (i)μt ∗ μs = μt+s, (ii)μ0 = limt→0+ μt = δ0,

where δ0 is the Dirac delta measure concentrated at 0 ∈ Rd and the limit exists

in the sense that for every f ∈ Cc(Rd), limt→0+

∫Rd

μt(dx)f(x) = f(0).

In the above, the convolution μ∗ν of two finite measures μ and ν is defined

as the unique finite measure such that∫f(x)(μ ∗ ν)(dx) =

∫ ∫f(x+ y)μ(dx)ν(dy), for all f ∈ C∞

c (Rd),

or equivalently, for a Borel set � ⊂ R, (μ ∗ ν)(�) =

∫μ(�− y)ν(dy). If μ and

ν are probability measures, then so is their convolution.

The next theorem gives the basic structure of such objects, justifying

semigroup in its name.

Theorem 2.5.7. Let {μt}t≥0 be a convolution semigroup on Rd and let 1 ≤ p <

∞. Define a map T (t) on Lp(Rd) by setting

(T (t)f)(x) =

∫f(x− y)μt(dy) and T (0)f = f for all f ∈ Lp(Rd) and x ∈ Rd.


Then {T (t)}t∈R+ is a C0 contraction semigroup on Lp(Rd), and is positive, that

is, (T (t)f)(x) ≥ 0 if f(x) ≥ 0, almost everywhere for all t ∈ R+. Furthermore,

T (t) commutes with the group of translations on Lp(Rd) and if f ∈ L1+(R

d)

(the positive elements of L1(Rd)), then ‖T (t)f‖1 = ‖f‖1.Proof. Since

|(T (t)f)(x)|p ≤(∫

Rd

|f(x− y)|μt(dy))p

and since [0,∞) � λ �→ λp is a convex function for each p ∈ [1,∞), we have by

Jensen’s inequality [23, Jensen’s Inequality, page 133] that

∣∣(T (t)f)(x)∣∣p ≤ ∫|f(x− y)|p μt(dy)

and therefore

‖T (t)f‖pp ≤∫

dx

∫|f(x− y)|p μt(dy).

An application of Fubini’s theorem and the observation that the Lebesgue

measure is translation invariant leads to the result that

∥∥T (t)f∥∥pp≤∫ ( ∫

|f(x− y)|p dx)μt(dy) = ‖f‖pp.

This means that T (t) is a contraction on Lp(Rd). Next, for t, s > 0, f ∈ Lp(Rd),

(T (t)T (s)f)(x) =

∫(T (s)f)(x− y)μt(dy) =

∫ (∫f(x− y − z)μs(dz)

)μt(dy)

=

∫f(x− y′)

∫μs(d(y

′ − y))μt(dy) =

∫f(x− y)(μs ∗ μt)(dy),

where we have made use of Fubini’s theorem to interchange the order of inte-

gration, made a change of variable and used the definition of the convolution

of two measures μs and μt. Thus we get that

(T (t)T (s)f)(x) =

∫f(x− y)μt+s(dy) = (T (t+ s)f)(x),

proving the semigroup property. Next we show strong continuity. Noting that

(T (t)f − f)(x) =

∫[f(x− y)− f(x)]μt(dy)

and applying Jensen’s inequality [23, Jensen’s Inequality, page 133], the above

leads to ∥∥T (t)f − f∥∥pp≤∫

dx

∫|f(x− y)− f(x)|pμt(dy). (2.27)

46 C0-semigroups

We now restrict f to be in C∞c (Rd). Recall that μ0 = limt→0+ μt = δ0 so that

limt→0+ μt{x ∈ Rd : |x| < α} = 1 for all α > 0, and therefore observe that

for small positive t, both the y- and x-integrals are effectively over bounded

subsets of Rd over which f is bounded. Given ε > 0, choose β > 0 such that

|f(x − y)− f(x)| < ε for |y| < β, for x in such bounded sets, where |y| is the

Euclidean norm of y ∈ Rd. Then choose t0 > 0 such that μt(|y| > β) < ε for

0 < t < t0. Thus,∫dx

∫|f(x− y)− f(x)|p μt(dy) =

∫dx

∫|y|<β

|f(x− y)− f(x)|p μt(dy)

+

∫dx

∫|y|>β

|f(x− y)− f(x)|p μt(dy)

= I1 + I2.

Now

I1 =

∫x∈(supp f+β)

dx

∫|y|<β

|f(x− y)− f(x)|pμt(dy)

< εpm(supp f + β),

where we have overestimated by using∫|y|<β μt(dy) ≤ 1. For the second integral

I2, we have

I2 ≤ 2p‖f‖ppμt(|y| > β) ≤ ε2p‖f‖pp.Combining these two estimates and (2.27), we get that

‖T (t)f − f‖p → 0, as t→ 0+ for any f ∈ C∞c (Rd).

Since C∞c (Rd) is dense in Lp(Rd) for 1 ≤ p < ∞, and ‖T (t)‖ is uniformly

bounded, we are able to extend the said convergence to the whole of Lp(Rd),

showing that {T (t)}t≥0 is a C0-semigroup. That {T (t)}t≥0 is positive is clear

from the definition and the fact that the μt’s are probability measures. Let

(S(a)f)(x) = f(x − a) for all a ∈ Rd, x ∈ Rd and f ∈ Lp(Rd), so that

{S(a)}a∈Rd is the group of translations. In fact, {S(a)}a∈R is a C0-group of

isometries on Lp(Rd) (compare with Remark 2.5.2 (2)). One notes that

(S(a)T (t)f)(x) = (T (t)f)(x− a) =

∫f(x− a− y)μt(dy)

=

∫(S(a)f)(x− y)μt(dy) = (T (t)S(a)f)(x). (2.28)


For f ∈ L1+(R

d), using Fubini’s theorem, we get that

‖T (t)f‖1 =∫(T (t)f)(x) dx =

∫dx

∫f(x− y)μt(dy)

=

∫μt(dy)

∫f(x− y) dx = ‖f‖1.

�

The most well-known example of a convolution semigroup is the heat

semigroup given by the Normal (0, t) probability measure on Rd :

μt(dx) = (2πt)−d/2e−|x|2

2t dx, (2.29)

where x ∈ Rd, |x| is the Euclidean norm of x and dx is the Lebesgue measure on

Rd. Similarly, the Cauchy measure also is of importance in probability theory:

on R,

γt(dx) = π−1t(t2 + x2)−1 dx.

It is left as an exercise (Exercise 2.5.8) to show that

(i) the heat and Cauchy measures satisfy the convolution semigroup proper-

ties of measures, as given in Definition 2.5.6, and

(ii) that the proof of Theorem 2.5.7 holds with Lp(R) replaced by C0(R), the

Banach space with supremum norm of continuous functions on R which

vanish at infinity.

Theorem 2.5.9. (Computation of the generator for the heat semigroup in

Lp(R).) The generator of the heat semigroup, associated with the convolu-

tion semigroup given by μt as in (2.29) is as follows:

D(A) = {f ∈ Lp(Rd) :

d∑i=1

∂2f

∂xj2∈ Lp(Rd)

}= W 2,p(Rd),

Af =1

2Δf =

1

2

d∑j=1

∂2f

∂xj2, for all f ∈ D(A),

that is, the generator of the heat semigroup is the Laplacian with maximal

domain, multiplied by 1/2 (see Section A.3).

Proof. For simplicity we will work with d = 1. Fix p ∈ [1,∞). From Theorem

2.5.7 we know already that the heat semigroup associated with the convolution

48 C0-semigroups

semigroup given by (2.29) is a C0 contraction semigroup on Lp(R) and hence, by

Theorem 2.2.7, its infinitesimal generator A is a densely defined closed operator

in Lp(R). For any f ∈ Lp(R), almost all x ∈ R and any t > 0,

(T (t)f − f)(x) =

∫R

(f(x− y)− f(x))(2πt)−1/2e−y2/2t dy

=

∫R

(f(x−√tu)− f(x))(2π)−1/2e−u2/2 du

=

∫R

(f(x−√tu)− f(x))N(u) du, (2.30)

where we have made a change of variable y =√tu, and have set N(u) =

(2π)−1/2e−u2/2 as the standard normal distribution function. Formally, we can

expand the expression in the parenthesis in (2.30) to get, for a sufficiently

smooth function f, that

f(x−√tu)− f(x) = −(√tu)f ′(x) +1

2(tu2)f ′′(x) +O(t3/2), (2.31)

as t→ 0+. Therefore,

(T (t)f − f)(x) = t/2f ′′(x) +O(t3/2),

since∫uN(u) du = 0 and

∫u2N(u) du = 1. Now we make this formal argument

rigorous. Let f ∈ D( d2

dx2 ) ⊂ Lp(R) (see Appendix A.3). Then proceeding as

above, we have

t−1(T (t)f − f)(x)− 1

2f ′′(x)

=

∫R

N(u) du t−1[f(x−√tu)− f(x)− t

2f ′′(x)

]=

∫R

N(u) du t−1[f(x−√tu)− f(x) +

√tuf ′(x)− t

2u2f ′′(x)

]=

∫R

N(u) du t−1

∫ −u√t

0

dα

∫ α

0

dβ[f ′′(x + β)− f ′′(x)

],

where we used the fact that f ′′ ∈ L1loc(R), the space of locally integrable func-

tions defined on R, and hence the fundamental theorem of integral calculus is

valid for it. Next, using the fact that R+ � λ �→ λp is a convex function for


p ≥ 1 and also using Jensen’s inequality twice in succession, we get that

∥∥t−1(T (t)f − f)− 1

2f ′′∥∥p

p

=

∫R

dx∣∣∣ ∫

R

N(u) du t−1

∫ −u√t

0

dα

∫ α

0

dβ[f ′′(x+ β)− f ′′(x)

]∣∣∣p≤∫R

N(u) du

∫dx∣∣∣t−1

∫ −√tu

0

dα

∫ α

0

dβ[f ′′(x+ β) − f ′′(x)

]∣∣∣p≤∫R

N(u) du t−p(12tu2

)(p−1)∫ |u|√t

0

dα

∫ α

0

dβ

∫R

∣∣f ′′(x + β)− f ′′(x)∣∣p dx

=

∫|u|>δt−1/2

N(u) du t−1(12u2)(p−1)

∫ |u|√t

0

dα

∫ α

0

dβ∥∥f ′′(·+ β)− f ′′(·)∥∥p

p

+

∫|u|<δt−1/2

N(u) du t−1(12u2)(p−1)

∫ |u|√t

0

dα

∫ α

0

dβ∥∥(S(β) − I)f ′′∥∥p

p

= I1 + I2,

where δ > 0. We have used the fact that the simplex 0 ≤ β ≤ α ≤ u√t has

the total area = (u2t/2) and set {S(β)}β∈R as the C0-group of isometries of

translation in Lp(R). Next, for any ε > 0,

I1 ≤ 2p‖f ′′‖pp∫|u|>δt−1/2

(u2

2

)pN(u) du

which we can make less than ε, by choosing t0 > 0 such that for 0 < t < t0, the

integral in I1 is less thanε

2p‖f ′′‖pp which is possible since

∫R

(u2

2

)pN(u) du <

∞.On the other hand, for I2 we note that S(β), the translation group on Lp(R),

is strongly continuous and hence uniformly strongly continuous for 0 ≤ β ≤ 1,

that is,

sup0≤β≤α≤|u|√t≤δ

‖(S(β)− I)f ′′‖pp < ε for 0 < t < t0.

Therefore,

I2 ≤ ε

∫|u|<δt−1/2

N(u)(u2

2

)(p−1)

t−1( tu2

2

)du < ε

∫N(u)

(u2

2

)pdu

showing that 12

d2

dx2 ⊂ A. For the other inclusion, that is, A ⊂ 12

d2

dx2 , we start

with the resolvent formula

(z −A)−1f =

∫ ∞

0

e−ztT (t)f dt, for z ∈ C such that Rez > ω, (2.32)

50 C0-semigroups

where ω is the exponential growth bound of T (t). Since T (t) is a contraction,

ω = 0, and we may choose z = 1. Furthermore, since the heat semigroup is

given by the integral kernel

K(t;x− y) = (2πt)−1/2e−(x−y)2

2t

so that

(T (t)f)(x) =

∫K(t;x− y)f(y) dy,

it is easily verified that the resolvent (1 − A)−1 is also an integral operator

given as

((1−A)−1f)(x) =

∫dy[ ∫ ∞

0

e−tK(t;x− y) dt]f(y). (2.33)

Using the fact that

∫ ∞

0

(2πt)−1/2e−(t+ x2

2t ) dt = 2−1/2e−√2|x| (see Lemma

2.5.10), and the above form of the resolvent, we have that

((1−A)−1f)(x) = 2−1/2

∫e−

√2|x−y|f(y) dy (2.34)

= 2−1/2

∫ x

−∞e−

√2(x−y)f(y) dy + 2−1/2

∫ ∞

x

e√2(x−y)f(y) dy

(2.35)

where both the integrals converge absolutely for each x ∈ R, by an application

of the Holder inequality along with the observation that f ∈ Lp(R) ⊂ L1loc(R).

Since (1 − A)−1 maps Lp(R) onto D(A) we conclude from (2.35) that every

element g ∈ D(A) is absolutely continuous and hence differentiable almost

everywhere. Furthermore, by differentiating on both sides of (2.35), we get

that for every f ∈ Lp(R) and for almost all x ∈ R,

((1−A)−1f)′(x) = −∫ x

−∞e−

√2(x−y)f(y) dy +

∫ ∞

x

e√2(x−y)f(y) dy

which is again absolutely continuous and is differentiable almost everywhere

and

((1 −A)−1f)′′(x) = 2[((1−A)−1f)(x) − f(x)]. (2.36)

The equation (2.36) establishes the facts that (i) every vector g ∈ D(A) ⊂Lp(R) is twice differentiable almost everywhere and (ii) g′′ ∈ Lp(R). Thus,

D(A) ⊂ D( d2

dx2 ), leading to the equality of the two domains and that A = 12

d2

dx2

on that domain.

�


Next we prove the lemma that was used in the proof in the preceding

paragraph.

Lemma 2.5.10. For x ∈ R,∫ ∞

0

(2πt)−1/2e−(t+x2

2t ) dt = 2−1/2e−√2|x|.

Proof. We begin with the well-known Gamma integral, viz,∫ ∞

0

e−y2

dy =

√π

2

and substitute y = α− c/α with c > 0 and√c ≤ α <∞. Thus,

√π

2=

∫ ∞

√c

e−(α−c/α)2(1 +

c

α2

)dα

= e2c∫ ∞

√c

e−(α2+ c2

α2

)dα+ e2c

∫ ∞

√c

e−(α2+ c2

α2 )( c

α2

)dα.

In the second integral on the right hand side another change of variable σ =

cα−1 converts the said integral into

−∫ 0

√c

e−(σ2+ c2

σ2

)dσ

and combining all these together we get that∫ ∞

0

e−(σ2+ c2

σ2

)dσ = e−2c

√π

2.

Finally, in the given integral in the statement, we set t = σ2 to get that∫ ∞

0

(2πt)−1/2e−(

x2

2t +t)dt =

2√2π

∫ ∞

0

e−(σ2+ x2

2σ2

)dσ = 2−1/2e−

√2|x|.

�

Chapter 3

Dissipative operators and

holomorphic semigroups

In this chapter we continue the study of C0-semigroups concentrating on con-

tractive and holomorphic semigroups. A brief summary of the frequently used

concepts and properties of densely defined closed or closable linear operators

in a Banach or Hilbert space can be found in Appendix A.1.

3.1 Dissipative operators

The Hille Yosida Theorem 2.3.1 and another version of it in Theorem 2.3.3

gives a characterisation for the generator of a contraction semigroup in terms

of certain resolvent estimates. Another, very different characterisation for such

generators is available via the notion of dissipative operators. We first define

dissipative operators on Hilbert spaces and then generalise to Banach spaces

before presenting the Lumer-Phillips characterisation, the primary goal of this

section.

Definition 3.1.1. Let X be a Hilbert space. A linear operator A on X is said

to be dissipative if Re 〈u,Au〉 ≤ 0 for all u ∈ D(A). If Re 〈u,Au〉 ≥ 0, then A

is called accretive.

Definition 3.1.2. A dissipative operator A on a Hilbert space X is called max-

imal dissipative if A does not admit any proper, dissipative extension.


54 Dissipative operators and holomorphic semigroups

Example 3.1.3. Let X = L2(0, 1) and A be the operator given by

D(A) = {f ∈ W 1,2(0, 1) : f(0) = 0}Af = f ′.

( For definition of W 1,2(0, 1) see Appendix A.3.) Then, for f ∈ D(A), |ff ′| ∈L1[0, 1] and f is absolutely continuous. Therefore,

Re 〈f,Af〉 = Re

∫ 1

0

f(t)f ′(t) dt =1

2

∫ 1

0

(f(t)f ′(t) + f(t)f ′(t)) dt

=1

2

∫ 1

0

d

dt(f(t)f(t)) dt =

1

2|f(1)|2 ≥ 0.

Thus, A is accretive and −A is a dissipative operator.

The proof of the following useful lemma is very simple and we leave the

details to the reader (Exercise 3.1.4).

Lemma 3.1.5. An operator A on a Hilbert space is dissipative if and only if

‖(A+ I)f‖ ≤ ‖(A− I)f‖ for all f ∈ D(A).

Theorem 3.1.6. For an operator A on a Hilbert space, the following are equiv-

alent:

(a) A is dissipative;

(b) ‖(A− λ)f‖ ≥ Reλ‖f‖ for all f ∈ D(A) and λ ∈ C with Reλ > 0;

(c) ‖(A− λ)f‖ ≥ λ‖f‖ for all f ∈ D(A) and λ > 0.

Proof. (a) ⇒ (b): For f ∈ D(A), we have, for Reλ > 0,

Re 〈f, (A− λ)f〉 = Re 〈f,Af〉 − Reλ‖f‖2 ≤ −Reλ‖f‖2 < 0.

Therefore,∥∥(A− λ)f∥∥‖f‖ ≥ ∣∣Re ⟨f, (A− λ)f

⟩∣∣ = −Re⟨f, (A− λ)f

⟩ ≥ Reλ‖f‖2.

(b) ⇒ (c): This is obvious.

(c) ⇒ (a): Let λ > 0 and f ∈ D(A). Since

‖Af‖2 − 2λRe 〈f,Af〉 = ‖Af − λf‖2 − λ2‖f‖2 ≥ 0,

3.1. Dissipative operators 55

we have 2λRe 〈f,Af〉 ≤ ‖Af‖2. But λ > 0 is arbitrary, so it follows that

Re 〈f,Af〉 ≤ 0.

�

Remark 3.1.7. As a direct consequence of Theorem 3.1.6, it follows that for

every λ with Reλ > 0, Ran(A − λ) is a closed subspace of H whenever A

is a closed, dissipative operator on H. Furthermore, for a dissipative operator

A, (A − λ) is injective for Reλ > 0, but may not be surjective. Suppose that

A is a closed, densely defined dissipative operator on a Hilbert space. Let λ

be such that Reλ > 0 and fn ∈ D(A) be such that (A − λ)fn is a Cauchy

sequence in H . From (b) of Theorem 3.1.6 it follows that (fn) is Cauchy and

so there exists f ∈ H with fn → f as n → ∞. But closedness of A implies

that f ∈ D(A) and (A − λ)fn → (A − λ)f as n → ∞. This implies that for

Reλ > 0, the Ran(A− λ) is a closed subspace of H. Furthermore,∥∥(A− λ)−1g∥∥ ≤ (Reλ)−1‖g‖, ∀g ∈ Ran(A− λ). (3.1)

But Ran(A − λ) is not necessarily all of H. As an example one can look at

the operator A0 in L2[0, 1] appearing in Example A.1.10. The domain of A∗0 is

easily seen to be

D(A∗0) = { f ∈ L2[0, 1] : f absolutely continuous, f ′ ∈ L2[0, 1] },

and therefore any vector orthogonal to Ran(A0 − λ) for some λ > 0 will be

the solution in D(A∗0) of the equation A∗

0f = λf. This equation can be seen

to be a classical differential equation and has a solution f(t) = ceiλt, where

c is a constant. This f ∈ L2[0, 1] and therefore, Ran(A0 − λ) �= L2[0, 1]. The

verification of these statements is left as an exercise (Exercise 3.1.8).

The property of being dissipative is stable under closure. This is made

precise in the following lemma.

Lemma 3.1.9. Any densely defined dissipative operator on a Hilbert space is

closable. The closure of A is again dissipative. Thus, a maximal dissipative

operator which is densely defined is closed.

Proof. Let A be a densely defined dissipative operator on the Hilbert space H

with domain D(A). Let {un}n ⊂ D(A) be a sequence converging to 0 such that

Aun → v as n→∞. For any u ∈ D(A) and α ∈ C, we have

Re 〈u+ αun, A(u + αun)〉 ≤ 0.


Letting n→∞ in the above gives

Re 〈u,Au〉+ Reα〈u, v〉 ≤ 0.

Since α is arbitrary, it follows that 〈u, v〉 = 0 for any u ∈ D(A). The density of

D(A) implies therefore that v = 0. Thus A is closable. Now A, the closure of

A is given by setting

D(A) = {u ∈ H : if {un} ⊂ D(A) with limn

un = u, then

there exists v ∈ H such that limn

Aun = v},

Au = v.

Then Re 〈u,Au〉 = limn Re 〈un, Aun〉 ≤ 0, for all u ∈ D(A). Thus A is also

dissipative. The second part of the lemma now follows from the property of

maximality.

�

Lemma 3.1.10. If A is a densely defined, dissipative operator on a Hilbert

space H, and Ran(A− λ) = H, for some λ with Reλ > 0, then A is maximal

dissipative.

Proof. Let A be densely defined and dissipative, with Ran(A − λ) = H for

some Reλ > 0. Let A be a dissipative extension of A, let u ∈ D(A) and let

v = (A− λ)u. Then there exists w ∈ D(A) such that

v = (A− λ)u = (A− λ)w = (A− λ)w.

On the other hand, A being dissipative implies (A − λ) is injective, so that

u = w. Thus u ∈ D(A) and A = A. �

Lemma 3.1.11. Every densely defined, dissipative operator on a Hilbert space

admits a maximal dissipative extension.

Proof. In view of Lemma 3.1.9 we may assume, without loss of generality, that

A is a closed, densely defined dissipative operator. Let λ be such that Reλ > 0

and set N = (Ran(A− λ))⊥. Then for v ∈ N and u ∈ D(A),

〈v,Au〉 = 〈v, (A − λ)u〉+ λ〈v, u〉.

Since 〈v, (A − λ)u〉 = 0, it follows that∣∣〈v,Au〉∣∣ ≤ |λ|‖v‖‖u‖. Therefore, v ∈

D(A∗) and A∗v = λv. If v ∈ D(A) ∩ N , then 〈v, (A − λ)v〉 = 0 which implies


that Reλ‖v‖2 = Re 〈v,Av〉 ≤ 0. Since Reλ > 0, it follows that v = 0. Thus,

D(A) ∩ N = {0}. Set

D :={u+ v : u ∈ D(A), v ∈ N}

,

A(u+ v) := Au− λv.

Then A is a linear operator with D(A) = D and A ⊂ A. We will now show that

A is a maximal dissipative extension of A. Let u ∈ D(A) and v ∈ N . Then

Re 〈u + v, A(u+ v)〉 = Re 〈u+ v,Au− λv〉= Re 〈u,Au〉+ Re 〈v,Au〉 − Reλ〈u, v〉 − Reλ‖v‖2

= Re 〈u,Au〉+ Re 〈(A− λ)u, v〉 − Reλ‖v‖2

= Re 〈u,Au〉 − Reλ‖v‖2

≤ 0.

Thus A is a dissipative extension of A. Since A is densely defined, so is A. In

view of Lemma 3.1.10, it is now sufficient to show that Ran (A − λ) = H.

As discussed in Remark 3.1.7, the facts that A is closed, densely defined and

dissipative ensures that Ran (A−λ) is a closed subspace of H. Thus any x ∈ H

is expressible uniquely as x = w + v, where w ∈ Ran (A− λ) and v ∈ N . This

implies that x = (A−λ)u+ v for some u ∈ D(A). Assuming that v �= 0, we set

y = u+ v1, where v1 = −(λ+ λ)−1v ∈ N . Then y ∈ D(A) and

(A− λ)y = (A− λ)(u + v1) = (A− λ)u − (λ+ λ)v1 = x.

Thus x ∈ Ran (A− λ).

�

Lemma 3.1.12. The following three statements are equivalent for a densely

defined, dissipative operator A on a Hilbert space H.

(i) A is maximal dissipative.

(ii) Ran(A− λ) = H for all λ with Reλ > 0.

(iii) Ran(A− λ) = H for some λ with Reλ > 0.

Proof. (i) ⇒ (ii): Let λ ∈ C be such that Reλ > 0. Assume that Ran (A− λ)

is not the whole of H. Then Ran(A−λ) is a proper, closed subspace of H. Set


N = (Ran (A−λ))⊥ �= {0}. Define A as in the proof of Lemma 3.1.11. This A

is then a dissipative extension of A, contradicting (i).

(ii) ⇒ (iii): This is trivial.

(iii) ⇒ (i): This follows from Lemma 3.1.10. �

Theorem 3.1.13. A densely defined operator A on a Hilbert space H is maximal

dissipative if and only if it is closed, {z ∈ C : Re z > 0} ⊂ ρ(A) and∥∥(A− z)−1∥∥ ≤ (Re z)−1 whenever Re z > 0.

Proof. Let A be a densely defined maximal dissipative operator. By Theorem

3.1.9, A is closed. Now A− z is injective since A is dissipative and by Theorem

3.1.12, A − z is also surjective for Re z > 0. The claim then follows from

Theorem 3.1.6 and (3.1) of Remark 3.1.7. Conversely, suppose that A is a

densely defined, closed operator satisfying {z ∈ C : Re z > 0} ⊂ ρ(A) and∥∥(A− z)−1∥∥ ≤ (Re z)−1 whenever Re z > 0. Again Theorem 3.1.6 shows that

A is dissipative and maximal dissipativity follows from Lemma 3.1.10.

�

Theorem 3.1.14. Let the densely defined closed operator A be dissipative. If

A is maximal dissipative, then A∗ is maximal dissipative. On the other hand,

if A∗ is dissipative, then A is maximal dissipative.

Proof. Suppose that A is a closed, densely defined maximal dissipative opera-

tor. Since A is closed, D(A∗) is dense. By Theorem 3.1.13, if Reλ > 0, then∥∥(A∗ − λ)−1∥∥ =

∥∥(A− λ)−1∥∥ ≤ (Reλ)−1.

In addition, we note that A∗ is a densely defined operator which is closed and

{λ : Reλ > 0} ⊂ ρ(A∗). Therefore, by Theorem 3.1.13 it follows that A∗

is maximal dissipative. Conversely, suppose A∗ is dissipative. Then it follows

from Theorem 3.1.6 that∥∥(A∗ − 1

)u∥∥ ≥ ‖u‖ for all u ∈ D(A∗). Suppose,

if possible that Ran (A − 1) �= H. But Ran (A − 1) is closed, since A is

dissipative, by Remark 3.1.7. Set N = Ran (A− 1)⊥ and let v ∈ N. Then, for

u ∈ D(A), 〈v, (A−1)u〉 = 0. This implies that v ∈ D(A∗) and 〈(A∗−1)v, u〉 = 0.

This is true for all u ∈ D(A). The density of D(A) implies that(A∗ − 1

)v = 0

so that v = 0. Thus N = {0}. So, Ran (A− 1) = H. The maximal dissipativity

of A now follows from Theorem 3.1.12.

�


Till now we have dealt with dissipative operators on Hilbert spaces. It is

possible to extend these ideas to Banach spaces as well. Let X be a Banach

space and recall that, as a consequence of the Hahn Banach Theorem, for any

x ∈ X, there exists an fx ∈ X∗ such that fx(x) = ‖x‖, and ‖fx‖ = 1. This

functional fx is not necessarily unique. Set

fx = ‖x‖fx.

Then fx(x) = ‖x‖2 = ‖fx‖2. This association x �→ fx from X → X∗ such that

fx(x) = ‖x‖2 = ‖fx‖2

is called a dual injection (sometimes also called a normalised tangent functional)

ofX inX∗. Using this association, we define dissipative operators in this general

setting:

Definition 3.1.15. An operator A on a Banach space X is said to be dissi-

pative if for every x ∈ D(A) there exists a dual injection fx of x such that

Re fx(Ax) ≤ 0. A is maximal dissipative if it does not admit any proper dissi-

pative extensions.

Here we note that if X is a Hilbert space then X∗ = X and fx = x

in this identification, so the dual injection is canonical and thus this general

definition gives back the one in Definition 3.1.1. The following theorem proves

properties in Banach spaces, similar to those in Hilbert spaces as proven earlier

in Theorem 3.1.6.

Theorem 3.1.16. The following are equivalent for an operator A on a Banach

space X.

(i) A is dissipative;

(ii) ‖(A− λ)u‖ ≥ Reλ‖u‖ for all u ∈ D(A) and Reλ > 0;

(iii) ‖(A− λ)u‖ ≥ λ‖u‖ for all u ∈ D(A) and λ > 0.

Proof. For u, v ∈ X, with ‖u‖ ≤ ‖u − αv‖ for all α > 0, we claim that there

exists fu ∈ X∗ such that Re fu(v) ≤ 0. Indeed, for u = 0, choose fu = 0.

If u �= 0, pick the dual injection fu−αv and denote it as fα for brevity. Set

gα = fα/∥∥fα∥∥. Then ‖gα‖ = 1 and by the w∗-compactness of the unit ball in


X∗, the net {gα}α>0 will have a convergent subnet, converging to g as α→ 0+

in w∗-topology with ‖g‖ ≤ 1. Thus,

‖u‖ ≤ ‖u− αv‖ = ‖u− αv‖2‖fα‖ =

fα(u − αv)

‖fα‖= gα(u − αv) = Re gα(u)− αRe gα(v)

≤ ‖u‖ − αRe gα(v), (3.2)

which implies that Re gα(v) ≤ 0, ∀ α > 0 and thus Re g(v) ≤ 0. On the other

hand, taking the limit as α → 0+ in (3.2) we get that Re g(u) ≥ ‖u‖ which

combined with the fact that Re g(u) ≤ |g(u)| ≤ ‖u‖ leads to the conclusion

that Re g(u) = ‖u‖. Set f = ‖u‖g ∈ X∗. Then f(u) = ‖u‖g(u) = ‖u‖2 = ‖f‖2.Thus f = fu and Re fu(v) = ‖u‖Re g(v) ≤ 0, thereby establishing the claim.

(iii) ⇒ (i): Suppose ‖(A−λ)u‖ ≥ λ‖u‖ for all λ > 0 and u ∈ D(A). Then∥∥u− λ−1Au

∥∥ = λ−1‖(A− λ)u‖ ≥ ‖u‖

or ‖u‖ ≤ ‖u−αAu‖ for all α > 0. From the first part of the proof, this implies

that there exists fu ∈ X∗ such that Re fu(Au) ≤ 0. Thus, A is dissipative.

(i) ⇒ (ii): Let u ∈ D(A) and Reλ > 0. Since A is dissipative, there

is a dual injection fu such that Re fu(Au) ≤ 0. Since Re fu((A − λ)u) =

Re fu(Au)− Reλfu(u) ≤ −Reλ‖u‖2, we have that

‖(A− λ)u‖‖u‖ = ‖(A− λ)u‖∥∥fu∥∥ ≥ ∣∣fu((A− λ)u)∣∣

≥ ∣∣Re fu(A− λ)u)∣∣ ≥ Reλ‖u‖2.

(ii) ⇒ (iii): This is obvious. �

As we have seen in the setting of a Hilbert space (Lemmas 3.1.11, 3.1.12

and Theorems 3.1.13 and 3.1.14) the density of the domain of a dissipative

operator is necessary to get any complete description of it. The same is true

in a Banach space as the next Theorem 3.1.18 will show. Before we prove the

theorem, we give an example of an operator that is dissipative, but not densely

defined.

Example 3.1.17. Let X = C[0, 1] and consider the operator

Af = −f ′, f ∈ D(A) = {g ∈ C1[0, 1] : g(0) = 0

}


Then A is a closed, dissipative operator on a Banach space which is not densely

defined, and (λ−A)D(A) = X, for all λ > 0. For if {fn} ⊂ D(A) is a sequence

converging to some f ∈ X and {Afn} converges to g, then this simply means

that {fn} is a sequence of functions converging uniformly to f, with {f ′n}

converging uniformly to g. This implies that f is also differentiable with f ′ = g

and f(0) = 0. Thus f ∈ D(A) and Af = g, showing that A is closed. Clearly,

D(A) �= X, as the constant function 1 ∈ X cannot be approximated by any

sequence in D(A). We now establish dissipativity of the operator A.

Let f ∈ D(A). For λ > 0 and g ∈ X,

(λ−A)f = λf + f ′ = g.

defines a classical linear ordinary differential equation. Solving the above dif-

ferential equation yields, for t ∈ [0, 1],

f(t) =

∫ t

0

e−λ(t−s)g(s) ds. (3.3)

Therefore,

|f(t)| ≤ ‖g‖e−λt

∫ t

0

esλ ds ≤ ‖g‖e−λt(etλ − 1

λ

),

where ‖g‖ = supt∈[0,1] |g(t)|.Thus, ‖f‖ ≤ ‖g‖/λ, or ‖(λ−A)f‖ ≥ λ‖f‖. Since this is true for all λ > 0

and f ∈ D(A), it follows from Theorem 3.1.16 that A is dissipative. Moreover,

for every g ∈ C[0, 1], the function f given by (3.3) is in D(A) and satisfies

λf + f ′ = g, that is, (λ− A)f = g. Therefore, Ran(λ−A) = X.

We are now in a position to prove the Lumer-Phillips Theorem [16] for

any Banach space.

Theorem 3.1.18 (Lumer-Phillips). Let A be a densely defined operator on a

Banach space X. Then A is the generator of a C0-semigroup of contractions if

and only if A is dissipative and Ran(A− 1) = X.

Proof. Suppose A is dissipative and Ran(A − 1) = X. Then, by Theorem

3.1.16(ii), 1 ∈ ρ(A),∥∥(A − 1)−1

∥∥ ≤ 1 and A is a closed operator (see Remark

3.1.19). If we show that λ ∈ ρ(A) for every λ > 0, and∥∥(A − λ)−1

∥∥ ≤ λ−1,

then by the Hille-Yosida Theorem 2.3.3 it will follow that A is the generator of


a C0 contraction semigroup. Let 0 < λ < 2 and set

R′ = (A− 1)−1[1− (λ− 1)(A− 1)−1

]−1 ∈ B(X),

where we have noted that the inverse of the term in the square parenthesis

exists in B(X) as the limit of a convergent Neumann series, since |λ − 1| < 1.

Then Ran R′ = D(A) and

(A− λ)R′ = (A− 1− (λ − 1))(A− 1)−1[1− (λ− 1)(A− 1)−1

]−1= I.

On the other hand, for u ∈ D(A),

R′(A− λ)u

= (A− 1)−1[1− (λ− 1)(A− 1)−1

]−1[1− (λ− 1)(A− 1)−1

](A− 1)u

= u.

This implies that (A − λ) is bijective and (A − λ)−1 = R′, so that λ ∈ ρ(A),

and by Theorem 3.1.16(iii), ∥∥(A− λ)−1∥∥ ≤ λ−1. (3.4)

Thus, (0, 2) ⊂ ρ(A) and every λ ∈ (0, 2) satisfies (3.4). Now fix λ0 ∈ (1, 2) and

let μ > 0 with |λ0 − μ| < ‖(A − λ0)−1‖−1. Proceeding as before, but with 1

replaced by λ0 we obtain that μ ∈ ρ(A) and ‖(A − μ)−1‖ ≤ μ−1. Continuing

in this way, we can cover all of (0,∞), that is, we obtain (0,∞) ⊂ ρ(A) and

that (3.4) holds for all λ ∈ (0,∞). Note that since ‖(A − λ0)−1‖−1 ≥ λ0 > 1

initially, all of (0,∞) can indeed be covered in the above mentioned manner.

Conversely, suppose {T (t)}t≥0 is a C0-semigroup of contractions with gen-

erator A. Let x ∈ X and let fx be a dual injection. Then

fx((T (t)− I)x) = fx(T (t)x)− fx(x) = fx(T (t)x)− ‖x‖2.

Therefore, since T is a contraction,

Re fx((T (t)− I)x) = Re fx(T (t)x)− ‖x‖2 ≤ ‖fx‖‖T (t)x‖ − ‖x‖2 ≤ 0.

If x ∈ D(A), limt→0+(T (t)x− x)/t = Ax. It follows therefore, that for any x ∈D(A), Re fx(Ax) ≤ 0. Finally, by Theorem 2.3.1, every z ∈ C with Re z > 0 is

in ρ(A) so that 1 ∈ ρ(A) and Ran(A− 1) = X.

�


Remark 3.1.19. Note that the statement of Theorem 3.1.18 does not mention

explicitly that A is a closed operator. However, it must be so in order to be

the generator of a C0-semigroup. And indeed the conditions of dissipativity

and surjectivity of (A−1) imply closedness of A. Let {un} ⊂ D(A) converge tou ∈ X and suppose that Aun → v as n→∞. Then (A−1)un → v−u as n→∞.

But since A is dissipative, it follows from Theorem 3.1.16 that ‖(A−I)u‖ ≥ ‖u‖for all u ∈ D(A). This, along with the hypothesis Ran (A − 1) = X implies

that 1 ∈ ρ(A), and (A − 1)−1 ∈ B(X). Therefore, un → (A − 1)−1(v − u) as

n→∞, or, u = (A− 1)−1(v − u) leading to the conclusion that u ∈ D(A) andAu = v.

Corollary 3.1.20. Let A be a densely defined closed linear operator in X, such

that both A and A∗ are dissipative. Then A is the generator of a C0 contraction

semigroup.

Proof. In view of Theorem 3.1.18 we just need to prove that Ran(A − 1) =

X. Since A is closed and dissipative, it follows that Ran(A − 1) is a closed

subspace of X. If Ran(A − 1) �= X, then there exists x∗ �= 0 in X∗ such

that 〈x∗, (A − I)x〉 = 0, for all x ∈ D(A). This implies that x∗ ∈ D(A∗) and

〈A∗x∗, x〉 = 〈x∗, x〉 for all x ∈ D(A). Density of D(A) implies that A∗x∗ = x∗.

Then

fx∗(A∗x∗) = ‖x∗‖2 = Re fx∗(A∗x∗) ≥ 0,

contradicting the dissipativity of A∗. �

Example 3.1.21. We revisit Example 3.1.3. As we saw earlier, the operator −Ais dissipative. We now show that the range condition of Theorem 3.1.18 also

holds. For f ∈ L2(0, 1) and λ > 0, set

u(t) =

∫ t

0

e−λ(t−s)f(s) ds.

Then u ∈ D(A) and λu+ u′ = f. Hence (λ+A)D(A) = X for all λ > 0. Also,

D(A) is dense in L2(0, 1). Thus, −A generates a contraction C0-semigroup on

X.

Example 3.1.22. Let X = L2(Rd), and set

D(A) = {f ∈ L2(Rd) : Δf ∈ L2(Rd)

}= H2(Rd),

Af = Δf,


where Δ is the Laplacian, defined in Appendix A.3. We shall show that A is a

densely defined, dissipative operator with (1−A)D(A) = X and hence conclude

from Theorem 3.1.18 that A generates a C0 contraction semigroup. Recall from

Appendix A.3 that Δf =∑d

i=1∂2f∂xj

2 , and that for f ∈ H2(Rd) ⊂ H1(Rd),

〈Δf, f〉L2(Rd) =

∫Rd

(Δf)f dx = −∫Rd

|∇f |2 dx ≤ 0.

Thus A is dissipative. Since C∞c (Rd) ⊂ H2(Rd), A is densely defined. We prove

next that Ran(1−A) = X. Let

φ(k) =1

|k|2 + 1and ψ(k) = |k|2 for all k ∈ Rd.

From Appendix A.2, one finds that (F(Δf))(k) = −ψ(k)(Ff)(k) and thus, if

for any g ∈ L2(Rd), we set u = F−1(MφFg), it follows from Lemma A.3.1 that

u ∈ H2(Rd) ⊂ L2(Rd). Furthermore, MψFu = MψMφFf. Therefore,

(1−Δ)u = F−1(MφFg) + F−1(MψFu) = F−1(MφFg) + F−1(MψMφFg)= F−1((I +Mψ)MφFg) = g.

Thus g ∈ Ran (1 − A) and hence X = Ran (1 − A). Therefore, by Theorem

3.1.18 it follows that A generates a C0-semigroup of contractions.

3.2 Stone’s Theorem

In this section, we obtain a classical theorem characterising the generator of

a unitary C0-group on a Hilbert space, as an application of Theorem 3.1.18.

By a unitary C0-group on a Hilbert space H, we mean a family {U(t)}t∈R of

unitary operators on H (that is, bounded operators on H satisfying U(t)∗ =

U(t)−1 ∀ t ∈ R) which forms a C0-group (see Definition 2.2.9). The group

property implies, in particular, that U(t)−1 = U(−t).Theorem 3.2.1. A is the generator of a C0 unitary group {U(t)}t∈R in a Hilbert

space H if and only if iA is selfadjoint.

Proof. Suppose first that A generates a C0-semigroup of unitaries, {U(t)}t∈R.

Then, setting

U+(t) := U(t) and U−(t) := U(−t) for all t ≥ 0,

3.2. Stone’s Theorem 65

we see that {U+(t)}t≥0 is a C0-semigroup of contractions, generated by A, while

{U−(t)}t≥0 is a C0-semigroup of contractions with generator −A. Furthermore,

since

t−1[U(t)∗x− x

]= t−1

[U(−t)x− x

], (3.5)

it follows, from Theorem 2.4.1 and the above discussion, on letting t→ 0 that

x ∈ D(A∗) if and only if x ∈ D(A), and in that case, A∗x = −Ax. Thus,(iA)∗ = iA, that is, iA is selfadjoint.

Conversely, assume that iA is selfadjoint, so that D(A)∗ = D(A), andA∗x = −Ax for all x ∈ D(A). Then

〈Ax, x〉 = 〈x,A∗x〉 = −〈x,Ax〉 = −〈Ax, x〉, ∀x ∈ D(A).

Thus Re 〈Ax, x〉 = 0 for all x ∈ D(A), so that both A and −A are dissipative.

Note that since A∗ is closed, being the generator of the C0-semigroup U−, A

is densely defined, that is, D(A) = H. Furthermore, A is closed since every

adjoint operator is necessarily closed (see Appendix A.1). By Theorem 3.1.18,

A generates a C0-semigroup, provided Ran(A− 1) = H. We establish this fact

next. From Remark 3.1.7 we have that (A−1) is injective, and that Ran(A−1)

is closed. Now suppose that g ⊥ Ran(A− 1). Then

〈g, (A− 1)f〉 = 0 ∀f ∈ D(A).

Thus, g ∈ D((A − 1)∗) = D(A∗) = D(A), and

−Ag − g = A∗g − g = (A− 1)∗g = 0.

Therefore,

Re 〈Ag, g〉 = 〈Ag, g〉 = −‖g‖2. (3.6)

Moreover,

〈Ag, g〉 = −〈A∗g, g〉 = −〈g,Ag〉 = −〈Ag, g〉.

Thus, Re 〈Ag, g〉 = 0, so that using (3.6), we have that g = 0. Thus, by

Theorem 3.1.18 it follows that A generates a C0-semigroup of contractions,

say, {U+(t)}t≥0. Starting with −A instead of A and using arguments similar

to above, we similarly conclude that −A also generates a C0-semigroup of

contractions,{U−(t)}t≥0. Since iA is selfadjoint, using the functional calculus


associated with selfadjoint operators (see [20, Theorem VIII.5, page 262 ]), we

can write

U+(t) = etA = e−it(iA), U−(t) = e−tA = eit(iA).

Define the family {U(t)}t∈R by

U(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩U+(t) if t > 0

I if t = 0

U−(t) if t < 0.

(3.7)

Using the above mentioned functional calculus one can check that the family

{U(t)}t∈R forms a group. Further, since U+ and U− are strongly continuous

and

limt→0+

U(t)x = limt→0+

(U+(t))x = x = lims→0+

(U−(s))x

= lims→0+

U(−s)x = limt→0−

U(t)x,

this family is strongly continuous on R. Moreover,

limt→0

t−1(U(t)x − x) = Ax, ∀x ∈ D(A).

Thus {U(t)}t∈R forms a C0-group with generator A. Since U(t)U(−t) = I =

U(−t)U(t) it follows that U(t)−1 = U(−t) for all t ∈ R . This implies that U(t)

is a unitary operator for each t. �

3.3 Holomorphic semigroups

Holomorphic semigroups play a very important role in the study of partial

differential equations. They are C0-semigroups with several special properties

and occur abundantly in applications. In fact, nearly all the semigroups that

we have seen thus far are holomorphic. Before introducing a formal definition,

we need to introduce the notion of holomorphy for vector-valued functions.

Definition 3.3.1. Let Ω be an open connected set in C, and let f : Ω → X,

where X is a Banach space.

1. f is said to be weakly holomorphic in Ω if the map Ω � z �→ 〈x∗, f(z)〉 isholomorphic for every x∗ ∈ X∗.

3.3. Holomorphic semigroups 67

2. f is said to be strongly holomorphic if the map Ω � z �→ f(z) ∈ X is

holomorphic in the strong (norm) topology of X.

It turns out that these two notions of holomorphy are actually equivalent,

by an application of Cauchy’s Theorem and of the Uniform Boundedness Prin-

ciple (Exercise 3.3.2). For the theory of vector-valued functions of a complex

variable, in particular the above mentioned property, the reader is referred to

[2, Appendix A].

We denote, for 0 < θ ≤ π, by Sθ, the sector

Sθ :={z ∈ C \ {0} : | arg z| < θ

}.

Definition 3.3.3. A holomorphic (or analytic) semigroup of angle θ, where

0 < θ ≤ π/2, defined on a Banach space X, is a family {T (z)}z∈Sθ∪{0} ⊂ B(X)

satisfying

(1) T (0) = I;

(2) the map z �→ T (z) is holomorphic in Sθ;

(3) T (z1)T (z2) = T (z1 + z2) for all z1, z2 ∈ Sθ;

(4) limSθ−ε�z→0 T (z)x = x, for all x ∈ X, whenever 0 < ε < θ.

If in addition,

(5) ‖T (z)‖ ≤Mε for all z ∈ Sθ−ε and for all ε such that θ > ε > 0,

then {T (z)}z∈Sθ∪{0} is called a bounded holomorphic semigroup.

It is clear that if {T (z)}z∈Sθ∪{0} is a holomorphic semigroup on X, then

{T (t)}t≥0 is a C0-semigroup on X. The generator A of this C0-semigroup is

referred to as the generator of the holomorphic semigroup {T (z)}z∈Sθ∪{0}. On

the other hand, a given C0-semigroup may not be extendable as a holomorphic

semigroup, as for example, a unitary group with its selfadjoint generator having

all of R as its spectrum.

The special properties of holomorphic semigroups are reflected in the gen-

erator of such a semigroup in several ways. Of these, perhaps the most impor-

tant ones are the shape of the resolvent set and the norm estimates of the

resolvent of the generator.

Theorem 3.3.4. Let {T (z)}z∈Sα∪{0} be a bounded holomorphic semigroup of

angle α, with generator A. Then {w ∈ C : | argw| < π2 + α} ⊂ ρ(A) and for


each ε ∈ (0, α), there is a constant Mε, such that

∥∥R(w,A)∥∥ ≤ Mε

|w| , ∀ w ∈ Sπ2 +α−ε.

Proof. Choose ε such that 0 < ε < α. Fix |θ| ≤ α − ε and set S(t) = T (teiθ)

for all t ≥ 0. Then, as {T (z)}z∈Sθ∪{0} satisfies the semigroup property,

S(t1)S(t2) = T (t1eiθ)T (t2e

iθ) = T ((t1 + t2)eiθ) = S(t1 + t2).

Further, S(0) = T (0) = I and S(t)x = T (teiθ)x → x as t → 0+. Therefore

(S(t))t≥0 is a C0-semigroup satisfying ‖S(t)‖ = ‖T (teiθ)‖ ≤ Mε, where Mε is

as in (5) of Definition 3.3.3. Let B be the generator of {S(t)}t≥0. Then for

x ∈ D(B) and s > 0, we have

S(s)Bx = limt→0+

S(s)t−1(S(t)x− x) = limt→0+

t−1(T ((t+ s)eiθ)x− T (seiθ)x)

= eiθ limt→0+

t−1(T (t+ seiθ)x− T (seiθ)x)

= eiθ limt→0+

t−1(T (t)− I)S(s)x = eiθAS(s)x.

This implies that S(s)D(B) ⊂ D(A) for any s > 0. As s → 0, the left

hand side of the above equation approaches Bx and, as A is closed, it fol-

lows that x ∈ D(A). In a similar manner, one can show that if x ∈ D(A), thenlims→0+ s−1(S(s)x − x) exists and equals eiθAx, so that

Bx = eiθAx ∀x ∈ D(B) = D(A). (3.8)

Since ‖S(s)‖ ≤Mε, it follows from Theorem 2.2.7(e) and (f) that

ρ(B) ⊃ {w : Rew > 0} and for such w,∥∥R(w,B)

∥∥ ≤ Mε

Re w.

Now, w ∈ ρ(A) if and only if w′ = weiθ ∈ ρ(B), and in such a case

∥∥(R(w,A)∥∥ =

∥∥R(w′, B)∥∥ ≤ Mε

Rew′ =Mε

(Reweiθ).

Therefore,

{e−iθλ : Re λ > 0} ⊂ ρ(A) and ‖(λeiθ −A)−1‖ ≤ Mε

Re λ, or{

rei(ψ−θ) : −π

2< ψ <

π

2

}⊂ ρ(A) and ‖(rei(ψ−θ) −A)−1‖ ≤ Mε

r cosψ, or{

w = reiγ : −π

2< γ + θ <

π

2

}⊂ ρ(A) and ‖(w −A)−1‖ ≤ Mε

|w| cos(γ + θ).


On the other hand, since |θ| ≤ α − ε, with 0 < α ≤ π/2, if −π2 < γ + θ < π

2 ,

then

−π

2− (α− ε) < γ + θ <

π

2+ (α− ε)

so that cos(argw + θ) ≥ sin(α− ε). Thus, it follows that{w : | argw| < π

2+ (α− ε)

} ⊂ ρ(A)

and, for all such w,

‖(w −A)−1‖ ≤ Mε

|w| sin(α− ε)=

M ′ε

|w| .

�

The conditions that the generator A in Theorem 3.3.4 above satisfies

makes it a sectorial operator of angle α. Precisely, we define sectorial operators

as follows.

Definition 3.3.5. A closed linear operator A with domain D(A) in a Banach

spaceX is called sectorial of angle δ where 0 < δ ≤ π2 , if the following conditions

hold.

1. The sector Sπ2+δ is contained in the resolvent set ρ(A).

2. For each ε ∈ (0, δ) there exists Mε ≥ 1 such that

‖R(λ,A)‖ ≤ Mε

|λ| ∀λ ∈ S π2 +δ−ε \ {0}.

The next theorem shows that every sectorial operator of angle less than

π/2 is the generator of a bounded holomorphic semigroup.

Theorem 3.3.6. Let A be a closed operator on X with Sπ2 +α ⊂ ρ(A), where

0 < α ≤ π/2, and such that for all ε > 0, there exists Nε > 0 satisfying

‖R(w,A)‖ ≤ Nε

|w| for all w ∈ Sπ2+α−ε.

Then the integral

T (z) :=1

2πi

∫Γ

ewzR(w,A) dw (3.9)

converges for a suitable smooth curve Γ in Sα+π2, for every z ∈ Sα. Moreover,

the family {T (z)}Sα∪{0} with T (0) = I, satisfies (2), (3), (4) and (5) of Def-

inition 3.3.3, thus forming a bounded holomorphic semigroup, with generator

A.


δ

Γ2

Γ3

Γ′

Γ1

Γ′

Γ′

θ

Figure 3.1: Contours Γ and Γ′

Proof. We first note that since the map w �→ ewzR(w,A) is holomorphic for

w ∈ Sα+π2, the integral in (3.9), if it exists, is independent of the choice of

Γ due to Cauchy’s Integral Theorem. We of course need to verify that the

contribution to the difference of the two integrals vanish in the limit as the

appropriate radius parameter increases to infinity (Exercise 3.3.7). We choose

Γ as follows.

Let ε be a suitably small positive number. Choose θ = π2 + α − ε/2. Let

Γ = Γ1 ∪ Γ2 ∪ Γ3 oriented anti-clockwise (see Figure 3.1) where

Γ1 ={re−iθ : δ < r <∞}

Γ2 ={δeiβ : −θ < β < θ

}Γ3 =

{reiθ : δ < r <∞}

and δ is a small positive number chosen so that 0 does not lie on Γ.


We first prove that the integral in (3.9) converges uniformly in B(X) for

z ∈ Sα−ε. Using the hypothesis on the resolvent of A, we have for z = reiψ

where |ψ| < α− ε, ∥∥ewzR(w,A)∥∥ ≤ eRe (wz)Nε

|w| . (3.10)

For w = reiθ ∈ Γ3 and z = |z|eiψ, Re (wz) = r|z| cos(θ+ψ). Since |ψ| < α− ε,

it follows that

π

2+ α− ε/2− (α− ε) < ψ + θ <

π

2+ α− ε/2 + α− ε;

and since 0 < α − ε < α < π2 we have that π

2 + ε/2 < ψ + θ < 3π/2 − ε/2.

Therefore, cos(θ + ψ) ≤ − sin ε/2 and

∥∥ewzR(w,A)∥∥ ≤ e−|w||z| sin(ε/2)Nε

|w| (3.11)

for all z ∈ Sα−ε and w ∈ Γ3. Similarly, it can be shown that (3.11) is true for

all z ∈ Sα−ε and w ∈ Γ1. For w ∈ Γ2 with δ = |z|−1 and z ∈ Sα−ε,∥∥ewzR(w,A)∥∥ ≤ e

Nε

|w| = eNε|z|. (3.12)

Therefore, for all z ∈ Sα−ε,

∥∥ 1

2πi

∫Γ

ewz(w −A)−1 dw∥∥ =

1

2π

3∑k=1

∥∥ ∫Γk

ewz(w −A)−1 dw∥∥

≤ 1

2π

( ∫Γ1

+

∫Γ3

)e−|w||z| sin(ε/2)Nε

|w| d|w|+ 1

2π

∫Γ2

eNε|z| d|w|

≤ Nε

π

∫ ∞

1/|z|

1

re−r|z| sin(ε/2) dr +

1

2πeNε|z|2π|z|−1

≤ Nε

π

∫ ∞

1

1

re−r sin ε

2 dr + eNε (3.13)

showing that the integral defining T (z) converges in B(X) absolutely and uni-

formly for all z ∈ Sα−ε. Since the above holds for every ε > 0, it follows that

the map z �→ T (z) is holomorphic on Sα, the details of the verification of which

is left as (Exercise 3.3.8). Moreover, it also follows that there exists a constant

Mε > 0 such that ‖T (z)‖ ≤ Mε ∀z ∈ Sα−ε. Next we check the semigroup

property. Let Γ′ be another contour, chosen as shown in Figure 3.1. Then as

noted in the beginning of this proof,

T (z) =1

2πi

∫Γ′

ewzR(w,A) dw, z ∈ Sα. (3.14)


Thus for z1, z2 ∈ Sα−ε, using the resolvent equation (Appendix A.1) we get

that

T (z1)T (z2) = (2πi)−2

∫w∈Γ

∫λ∈Γ′

ewz1eλz2R(w,A)R(λ,A) dw dλ

= (2πi)−2

∫w∈Γ

∫λ∈Γ′

ewz1+λz2(λ − w)−1[R(w,A) −R(λ,A)

]dw dλ

= (2πi)−2

∫w∈Γ

∫λ∈Γ′

ewz1+λz2(λ − w)−1R(w,A) dw dλ

− (2πi)−2

∫w∈Γ

∫λ∈Γ′

ewz1+λz2(λ− w)−1R(λ,A) dw dλ

= (2πi)−2(I1 + I2). (3.15)

Now by Cauchy’s Theorem, for λ ∈ Γ′,∫w∈Γ

ewz1(λ− w)−1dw = 0

since the completion of Γ does not enclose λ, and∫λ∈Γ′

eλz2(λ− w)−1 dλ = 2πiewz2.

These, together with Fubini’s Theorem, leads to:

I1 =

∫Γ

ewz1( ∫

Γ′

eλz2(λ− w)−1 dλ)R(w,A) dw

= 2πi

∫Γ

ewz1ewz2R(w,A) dw

= 2πi

∫Γ

ewz1+wz2R(w,A) dw

= (2πi)2T (z1 + z2),

while I2 = −∫w∈Γ

∫λ∈Γ′

ewz1+λz2(λ− w)−1R(λ,A) dw dλ

=

∫λ∈Γ′

eλz2( ∫

w∈Γ

ewz1(λ− w)−1 dw)R(λ,A) dλ

= 0.

It now follows from (3.15) that T (z1)T (z2) = T (z1+ z2). To complete the

proof, we need to show that the map z �→ T (z) is strongly continuous in Sα−ε


for every ε > 0. For x ∈ D(A) and z ∈ Sα−ε, we have that

T (z)x− x = (2πi)−1

∫Γ

ew[(w − zA)−1x− w−1x] dw

= (2πi)−1

∫Γ

w−1ew(w − zA)−1zAxdw.

Therefore, proceeding as in the estimation of the integrals in (3.13) with δ =

|z|−1 in Figure 3.1, we get that

‖T (z)x− x‖ ≤ 1

2π‖Ax‖

[2Nε

∫ ∞

|z|−1

r−2 dr + eNε|z|2|z|−12θ]

=Nε

π‖Ax‖(1 + eθ)|z| → 0 as |z| → 0.

Since A is densely defined and ‖T (z)‖ is uniformly bounded in Sα−ε it follows

that T (z)x − x → 0 for all x ∈ X. The continuity for all z ∈ Sα−ε is the

consequence of the already established semigroup law. �

Theorem 3.3.6 above shows that a given uniformly bounded C0-semigroup

{T (t)}t≥0 with generator A can be extended as a bounded holomorphic semi-

group on a sector containing the positive real axis provided A is sectorial (in

the sense of Definition 3.3.5) of angle less than π/2. Also note that intuitively,

the domain of holomorphy (in z) of a bounded holomorphic semigroup {T (z)}differs from the domain of holomorphy of the resolvent of its generator A or the

angle of sectoriality of A by π/2. We now look at other, different but related

conditions for a C0-semigroup to admit a holomorphic extension. We shall need

the following Lemma for this purpose.

Lemma 3.3.9. Let {T (t)}t≥0 be a C0-semigroup on a Banach space X with

generator A. Suppose that T (t)X ⊂ D(A) for every t > 0. Then, for each

x ∈ X, t �→ T (t)x is arbitrarily often (strongly) differentiable on R+ and

T (n)(t)x :=dn

dtnT (t)x =

([ d

dsT (s)

]s= t

n

)nx = AnT (t)x = [AT (t/n)]nx,

(3.16)

and AnT (t) ∈ B(X) for all t > 0, x ∈ X and n ∈ N.

Proof. Suppose t > t0 > 0. Then T (t)x = T (t − t0)T (t0)x. Since T (t0)x ∈D(A), using (b) of Theorem 2.2.7 we have that T ′(t)x exists and T ′(t)x =

T (t − t0)AT (t0)x = AT (t)x = T (t/2)AT (t/2)x ∈ D(A). Repeating the same


reasoning again yields that T ′′(t)x exists and that

T ′′(t)x = [AT (t/2)]2x = A2T (t)x, for every x ∈ X and t > 0.

The equality (3.16) follows now by induction (Exercise 3.3.10). Next we note

that for t > 0, the linear operator AT (t) is defined everywhere, that is,

D(AT (t)) = X, and since A is closed, it is easy to see that AT (t) is closed.

Therefore, by the Closed Graph Theorem, AT (t) ∈ B(X), for each t > 0. By

(3.16), it follows then that for t > 0, and n ∈ N,

AnT (t) = [AT (t/n)]n = T (n)(t) ∈ B(X).

�

Theorem 3.3.11. The following are equivalent for a uniformly bounded, C0-

semigroup {T (t)}t≥0 defined on a Banach space X, with generator A.

(i) For all t > 0, T (t)X ⊂ D(A), and there exists a constant M > 0 indepen-

dent of t such that∥∥tAT (t)∥∥ ≤M.

(ii) {T (t)}t≥0 admits an extension to a bounded holomorphic semigroup

{T (z)}Sα∪{0}, where tanα = 1Me , and for every α′, with 0 < α′ < α,

there is a constant Cα′ such that

‖T (z)‖ ≤ Cα′ ∀z ∈ Sα′ .

(iii) There exists a positive constant K > 0 such that for all a > 0 and b �= 0,∥∥R(a+ ib, A)∥∥ ≤ K

|b| . (3.17)

Proof. (i) ⇒ (ii): Since T (t)X ⊂ D(A), it follows from Lemma 3.3.9 that T (t)

is infinitely differentiable, and for all t > 0 and n = 1, 2, . . . ,( t

n

)n

T (n)(t) =( t

nT ′(t/n))n =

(t/nAT

(t/n

))n.

Hence ∥∥∥( t

n

)nT (n)(t)

∥∥∥ =∥∥∥( t

nAT

(t/n

))n∥∥∥ ≤Mn, (3.18)

by the hypothesis. Now (3.18) leads to the estimate that∥∥∥ (z − t)n

n!T (n)(t)

∥∥∥ =∥∥∥nn(z − t)n

tnn!

((t/n)nT (n)(t)

)n∥∥∥

≤ (nM)n

n!

∣∣∣z − t

t

∣∣∣n ≤ (Me)n∣∣∣z − t

t

∣∣∣n,


where we have used the inequalities

(n− 1)! = Γ(n) =

∫ ∞

0

e−xxn−1 dx >

∫ n

0

e−xxn−1 dx > e−nnn

n,

to get that nn

n! < en. Thus, the power series

∞∑k=0

(z − t)nT (n)(t)

n!(3.19)

converges in B(X) in operator norm, uniformly for all z ∈ C such that |z− t| <t/(Me). Setting

T (z) :=∞∑

n=0

(z − t)nT (n)(t)

n!(3.20)

we see that T (z) = T (t) for z = t ∈ R and that T (t) has a strong holo-

morphic extension T (z) to the sector Sα, where tanα = 1/eM. Since the

family {T (t)}t≥0 satisfies the semigroup property, the Identity Theorem for

holomorphic functions ([2, Proposition A.2]) ensures that so does {T (z)}z∈Sα.

Indeed, for fixed t > 0, the analytic map Sα � z → T (t)T (z) satisfies

T (t)T (z) = T (t+ z) for all z ∈ Sα ∩ R+. Therefore, by the Identity Theorem,

T (t)T (z) = T (t+ z) ∀ z ∈ Sα. Now, repeating this argument, with t replaced

by an arbitrary, but fixed z1 ∈ Sα yields finally T (z1)T (z2) = T (z1+ z2) for all

z1, z2 ∈ Sα.

Next we prove uniform boundedness. Let 0 < α′ < α and choose k ∈ (0, 1)

such that tanα′ = k/eM. For z = a+ ib ∈ Sα′ , we have, using the power series

for T (z) given in (3.20) and estimates obtained earlier in the proof,

‖T (a+ ib)‖ =∥∥∥T (a) + ∞∑

n=1

1

n!(ib)nT (n)(a)

∥∥∥≤ ‖T (a)‖+

∞∑n=1

|b|n(eM

a

)n

≤ ‖T (a)‖+∞∑

n=1

( k

eM

)n

(eM)n = ‖T (a)‖ − 1 +1

1− k, (3.21)

proving the uniform boundedness of {T (z)} in Sα′ . For x ∈ X, z ∈ Sα and

t > 0, we write, using the semigroup law that

T (z)x− x =[T (z + t)x− T (t)x

]− (T (z)− I)[T (t)x− x

].


This, along with the facts that {T (t)}t≥0 is a C0-semigroup and that the map

z → T (z + t)x is holomorphic in a neighbourhood of z = 0, and the estimate

(3.21) leads to the conclusion that T (z)x→ x as z → 0 in Sα for each x ∈ X

(Exercise 3.3.12).

(ii) ⇒ (iii): By Theorem 3.3.4, A is sectorial. In particular, (3.17) holds.

(iii)⇒ (i): We first show that if (3.17) holds, then A is sectorial in the sense

of Definition 3.3.5. Since {T (t)}t≥0 is uniformly bounded, by Theorem 2.2.7,

(e) and (f) we have that {λ ∈ C : Reλ > 0} ⊂ ρ(A) and for λ = a+ ib, a > 0,∥∥R(a+ ib, A)∥∥ ≤ C

a

for some constant C > 0. This estimate combined with (3.17) implies that∥∥R(λ,A)∥∥ ≤ K ′

|λ| for all λ such that Reλ > 0. (3.22)

Writing now the Taylor expansion for R(λ,A) (see (A.2)) around λ0 where

λ0 = a+ ib, a > 0, we have

R(λ,A) =∞∑

n=0

R(λ0, A)(n+1)(λ0 − λ)n. (3.23)

This series converges in B(X) for all λ ∈ C such that

‖R(λ0, A)‖|λ0 − λ| ≤ k < 1.

Letting λ = μ+ ib in above and on using the estimate

‖R(λ0, A)‖|λ0 − λ| = ‖R(λ0, A)‖|a− μ| ≤ K

|b| |a− μ|,

we have that the series in (3.23) converges uniformly in B(X) for all λ = μ+ ib

such that |a−μ| ≤ k|b|K . Thus λ = μ+ib ∈ ρ(A) for all λ satisfying |a−μ| ≤ k|b|

K .

Since a > 0 and k < 1 are arbitrary, it follows that

{λ ∈ C : Reλ ≤ 0, |Reλ|/|Imλ| < 1/K} ⊂ ρ(A).

In particular, this implies that Sα+π/2 ⊂ ρ(A), where tanα = 1/K. To obtain

the estimate on the resolvent in this region, we use the expansion in (3.23) to

get,

‖R(λ,A)‖ ≤∞∑n=0

‖R(λ0, A)‖(n+1)|λ0 − λ|n ≤ K

|b|∞∑

n=0

kn

=K

(1− k)|b| ≤√K2 + 1

(1− k)|λ| =M ′

|λ| .

3.4. Some examples of holomorphic semigroups 77

Thus A is sectorial.

Therefore, by Theorem 3.3.6, A generates a bounded holomorphic semi-

group {T (z)}z∈Sα, which extends the semigroup {T (t)}t≥0. In particular, the

map (0,∞) � t �→ T (t)x is differentiable for all x ∈ X, implying that the limit

lims→0+

s−1(T (t+ s)− T (t)

)x = lim

s→0+s−1

(T (s)− I

)T (t)x

exists for all x ∈ X and t > 0 so that T (t)X ⊂ D(A) for all t > 0. From the

proof of Theorem 3.3.6, with the same notations as in Figure 3.1, it follows by

using Lemma 1.2.5(ii) that for t > 0

AT (t) =1

2πiA

∫Γ

eλtR(λ,A) dλ =1

2πi

∫Γ

eλt(λR(λ,A) − I) dλ

=1

2πi

∫Γ

λeλtR(λ,A) dλ.

As in the proof of Theorem 3.3.6, we choose Γ2 such that δ = t−1. Thus, using

(3.11) and (3.12), we have

‖AT (t)‖ = ∥∥ 1

2πi

( ∫Γ1

+

∫Γ2

+

∫Γ3

)λeλtR(λ,A) dλ

∥∥≤ 1

2π

(2

∫ ∞

δ

Nεe−rt sin(ε/2) dr +

∫ θ

−θ

eNεδ dα)

≤ 1

2π

2Nε

t

( 1

sin(ε/2)+ πe

)=

M

t.

The proof is now complete. �

3.4 Some examples of holomorphic semigroups

Example 3.4.1. Consider the multiplication semigroup defined in Example 2.5.4

with generator Mq. This semigroup extends to a bounded holomorphic semi-

group of angle α if and only if

Sα+π/2 ⊂ ρ(Mq). (3.24)

Recall from (2.23) that σ(Mq) = ess ran q. From Theorem 3.3.4 it follows

that if Mq generates a bounded holomorphic semigroup of angle α then (3.24)

holds. To prove the converse, we invoke Theorem 3.3.6 and Theorem A.1.13 (i).


So suppose that Sα+π/2 ⊂ ρ(Mq) = C \ ess ran q. Let ε > 0 be arbitrary and

suitably small. For λ ∈ Sα+π/2−ε, we have, on using (2.23) and the discussion

preceding it, that,

‖R(λ,Mq)‖ = ‖Mψ‖ = ‖ψ‖∞ = sup{ 1

|q(s)− λ| : s ∈ ess ran q}

=1

dist(λ, ess ran q)≤ 1

dist(λ,C \ Sα+π/2)

≤ Nε

|λ|where ψ is as in (2.23) and Nε is a constant depending on α and ε. Since ε > 0

is arbitrary the claim follows from Theorem 3.3.6.

Example 3.4.2. Suppose A is a selfadjoint operator on a Hilbert space H and

there exists γ ∈ R, with γ ≤ 0, such that

〈Ax, x〉 ≤ γ〈x, x〉 ∀ x ∈ D(A). (3.25)

Then A is the generator of a bounded holomorphic semigroup of angle π/2.

By the Spectral Theorem A.1.15, we can assume that H = L2(Ω, μ) where

μ is a σ-finite measure, and that A = Mq, where q : Ω → C is a measurable

function, and Mq is the multiplication operator with respect to q. Since A

is selfadjoint, so is Mq and therefore, by Theorem A.1.13, q is real valued.

Moreover, (3.25) applied to Mq implies that

ess sups∈Ω Re q(s) ≤ γ.

Thus, by Theorem 2.5.4, Mq generates a C0-semigroup. Moreover, (3.25) im-

plies that σ(A) ≡ σ(Mq) ⊂ (−∞, 0]. Therefore condition (3.24) above holds

for Mq with α = π/2. We conclude from Example 3.4.1 that Mq and hence A

generates a bounded holomorphic semigroup of angle π/2.

Example 3.4.3. The heat semigroup discussed in Example 2.5.9, on X =

L2(Rd), extends to a bounded holomorphic semigroup of angle π/2.

The generator A of this semigroup is 12Δ with maximal domain H2(Rd)

(see Theorem 2.5.9). By Remark A.2.6(d) , (FΔf)(k) = −|k|2(Ff)(k) for all

f ∈ D(Δ), that is, Δ is unitarily equivalent to Mφ with φ(k) = −|k|2 and

hence Δ is a non-positive selfadjoint operator in L2(Rd) by Theorem A.1.13.

Therefore, A is also a non-positive selfadjoint operator. Therefore, it follows

3.4. Some examples of holomorphic semigroups 79

that A satisfies the conditions of Example 3.4.2 with γ = 0 and the heat

semigroup is bounded holomorphic of angle π/2.

Remark 3.4.4. The probability measure on Rd associated with the heat semi-

group as given in (2.29) makes sense for a complex parameter z replacing t :

μz(dx) = (2πz)−d/2 exp(− |x|

2

2z

)dx, (3.26)

with Re z > 0, and the branch of the square root is chosen such that Re√z > 0.

We can verify (Exercise 3.4.5) that the associated heat semigroup T (z), as de-

fined in Theorem 2.5.7 is well defined in Lp(Rd) (1 ≤ p <∞) as a holomorphic

semigroup of angle π/2. It is interesting to note that though the line Re z = 0

is not in the domain of holomorphy, {T (it)}t∈R does make sense as a unitary

group in L2(Rd), and as a family of bounded maps from L1(Rd) into L∞(Rd).

In fact, we can define

μit(dx) =

⎧⎨⎩(2π|t|)−d/2e−idπ/4 exp

( i|x|22t

)dx if t > 0,

(2π|t|)−d/2eidπ/4 exp( i|x|2

2t

)dx if t < 0,

and it can be shown that the associated semigroup has the above mentioned

properties (Exercise 3.4.6). The unitary group resulting from this exercise is

called the Schrodinger free evolution group and is of interest in Quantum Me-

chanics. For further reading in this area, the reader is referred to [1] and [21].

Chapter 4

Perturbation and convergence of

semigroups

In this chapter, the stability of various classes of semigroups under suitable sets

of perturbations will be studied, viz. for general C0-semigroups and contraction

semigroups. The methods involved will be the expansion of either the perturbed

semigroup itself in terms of the unperturbed one and the perturbation of the

generator or in terms of the resolvents concerned.

4.1 Perturbation of the generator of a C0-semigroup

Theorem 4.1.1. Let {T (t)}t≥0 be a C0-semigroup with generator A in a Banach

space X satisfying ‖T (t)‖ ≤ Meβt ∀ t≥ 0 where M ≥ 0 and β ∈ R, and let

B ∈ B(X). Then A + B is the generator of a C0-semigroup {S(t)}t≥0 with

bound ‖S(t)‖ ≤Me(β+M‖B‖)t.

Proof. For x ∈ X, t ≥ 0, set

S(t)x = T (t)x+

∞∑n=1

In(t)x, (4.1)

where

I0(t)x = T (t)x

I1(t)x =

∫ t

0

T (t− t1)BT (t1)x dt1


82 Perturbation and convergence of semigroups

I2(t)x =

∫ t

0

T (t− t2)BI1(t2)x dt2

=

∫ t

0

T (t− t2)B dt2

∫ t2

0

T (t2 − t1)BT (t1)x dt1

=

∫dt2

∫0≤t1≤t2≤t

dt1 T (t− t2)BT (t2 − t1)BT (t1)x,

and so on. Thus, In is recursively defined by setting

In(t)x =

∫ t

0

T (t− tn)BIn−1(tn)x dtn for all n ≥ 1.

First of all, we note that

‖In(t)x‖ ≤ (M‖B‖)n‖x‖Meβt∫0≤t1≤t2...≤tn≤t

dt1 dt2 . . . dtn

≤ ‖x‖ (M‖B‖t)n

n!Meβt, (4.2)

so that the infinite series

∞∑n=1

In(t) converges in operator norm (uniformly for t

in a compact interval) and defines S(t) as a bounded linear operator for every

t, with ‖S(t)‖ ≤Me(β+M‖B‖)t. Since In(0) = 0 ∀n ≥ 1, we have that S(0) = I,

and the strong continuity of S(t) follows from the same property of T (t) and

In(t), and the fact that∑∞

n=0 In(t) converges in operator norm uniformly in

compact intervals. Next, to show that {S(t)}t≥0 is a C0-semigroup, we first

need to establish the integral equation,

S(t)x = T (t)x+

∫ t

0

T (t− s)BS(s)x ds for all x ∈ X. (4.3)

This follows by rewriting the infinite sum on the right hand side of (4.1) as

S(t)x− T (t)x =

∞∑n=1

∫ t

0

T (t− s)BIn−1(s)x ds

=

∫ t

0

T (t− s)B( ∞∑n=1

In−1(s)x)ds

=

∫ t

0

T (t− s)B( ∞∑n=2

In−1(s)x+ T (s)x)ds

=

∫ t

0

T (t− s)BS(s)x ds,

4.1. Perturbation of the generator of a C0-semigroup 83

where the interchange of integration and summation above is justified by uni-

form convergence as seen before. Therefore,

S(t+ s)− S(t)S(s)

= T (t+ s) +

∫ t+s

0

T (t+ s− v)BS(v) dv

−(T (t) +

∫ t

0

T (t− u)BS(u) du)S(s)

= T (t+ s)− T (t)(T (s) +

∫ s

0

T (s− u)BS(u) du)

+(∫ t+s

0

T (t+ s− v)BS(v) dv −∫ t

0

T (t− u)BS(u)S(s) du)

= −∫ s

0

T (t+ s− u)BS(u) du+

∫ t+s

0

T (t+ s− v)BS(v) dv

−∫ t

0

T (t− u)BS(u)S(s) du

=

∫ t

0

T (t− u)B(S(u+ s)− S(u)S(s)) du.

On iterating the above n times we get the norm estimate

‖S(t+ s)− S(t)S(s)‖ ≤ (M‖B‖t)nn!

e|β|t supu∈[0,t]

(‖S(u)‖‖S(s)‖+ ‖S(u+ s)‖).

The left hand side is independent of n while the right hand side converges to 0

as n → ∞, showing that S satisfies the semigroup property. Now, using (4.3)

for x ∈ D(A), we have that

t−1(S(t)− I)x = t−1(T (t)− I)x+ t−1

∫ t

0

T (t− s)BS(s)x ds, (4.4)

and the right hand side of (4.4) converges strongly to Ax + Bx as t → 0+,

where we have applied Lemma 2.1.2 to show that the second term on the right

hand side of (4.4) converges to Bx as t → 0+. It is easy to see that A + B,

defined on D(A), is a closed operator since B is bounded. Therefore, it follows

that A+B is the generator of the semigroup {S(t)}t≥0.

�

The above result shows that perturbing the generator of a C0-semigroup

by a bounded operator again yields a generator of a C0-semigroup. The require-

ment of bounded perturbation can be weakened to relatively bounded perturba-

tion to obtain the same conclusion. Here we introduce the definition of relative


boundedness of operators and use it for proving results on perturbation of con-

traction semigroups, leaving further discussion about this topic to Section 4.2.

Definition 4.1.2. Let A : D(A) ⊂ X → X, be a linear operator on a Banach

space X . An operator B : D(B) ⊂ X → X is said to be bounded relative to A

if D(A) ⊂ D(B) and if there exist constants a, b ∈ R+ such that

‖Bx‖ ≤ a‖Ax‖+ b‖x‖ (4.5)

for all x ∈ D(A). The relative bound of B with respect to A is defined as

aA(B) := inf{a ≥ 0 : there exists b ∈ R+ such that (4.5) holds }. (4.6)

Theorem 4.1.3. Let A and B be generators of C0 contraction semigroups

{T (t)}t≥0 and {S(t)}t≥0 respectively in X and let B be bounded relative to A

with relative bound less than 12 . Then A+B is the generator of a C0 contraction

semigroup.

Proof. Let a and b be as in (4.5). Set x(t) = S(t)T (t)e−tλx with x ∈ D(A) ⊆D(B). Then for t > 0 and λ > 0,

t−1(x(t) − x) = S(t)t−1(e−tλT (t)x− x) + t−1(S(t)− I)x,

which converges strongly to (A− λ)x+Bx as t −→ 0+. On the other hand,

‖t−1(x(t)− x)‖ ≥ t−1(1− e−tλ)‖x‖ −→ λ‖x‖ as t −→ 0+

and therefore

‖[λ− (A+B)]x‖ ≥ λ‖x‖

for all x ∈ D(A) and λ > 0, which implies that λ− (A+B) is injective. Recall

that λ ∈ ρ(A) for all λ > 0, hence by Theorem 2.2.7

‖B(λ−A)−1‖ ≤ b‖(λ−A)−1‖+ a‖A(λ−A)−1‖≤ bλ−1 + a(1 + λ‖(λ−A)−1‖) ≤ bλ−1 + 2a < 1

for sufficiently large λ, since a < 1/2. This along with the equality

λ− (A+B) = (I −B(λ−A)−1)(λ−A),

4.1. Perturbation of the generator of a C0-semigroup 85

implies that Ran(λ− (A+B)) = X for sufficiently large λ, say for λ > λ0, and

hence, for such λ > λ0, ‖(λ − (A + B))−1‖ ≤ λ−1. Next, let 0 < μ < λ0 < λ

and note that

[μ− (A+B)] = [λ− (A+B)][1 + (μ− λ)(λ − (A+B))−1],

yielding that

[μ− (A+B)]−1

= [1 + (μ− λ)(λ − (A+B))−1]−1

[λ− (A+B)]−1

,

where the Neumann series for [I + (μ − λ)(λ − (A + B))−1]−1 converges in

operator norm since the series

∞∑n=0

( |μ−λ|λ

)nconverges. Thus

∥∥(μ− (A+B))−1∥∥ ≤ λ−1

∞∑n=0

( |μ− λ|λ

)n= μ−1,

proving the necessary estimate for all μ > 0. Furthermore, by Theorem 4.2.1(i)

A+B, defined on D(A) is closed. An application of The Hille-Yosida Theorem

2.3.3(i) leads to the desired result.

�

Corollary 4.1.4. In the previous theorem, it is sufficient to assume that a < 1

instead of < 12 .

Proof. Suppose for some α with 0 ≤ α < 1, A + αB is the generator of a C0

contraction semigroup. For this note that A + αB is defined on D(A) and is

closed by Theorem 4.2.1, since the relative bound a of B is less than 1. Then

(1− aα)‖Bx‖ ≤ a(‖Ax‖ − α‖Bx‖) + b‖x‖ ≤ a‖(A+ αB)x‖ + b‖x‖

or

‖Bx‖ ≤ a(1− aα)−1‖(A+ αB)x‖ + b(1− aα)−1‖x‖

and therefore, if we choose β with 0 ≤ β ≤ 12 (1− a), then βa(1− aα)−1 < 1

2

and we can apply Theorem 4.1.3 with βB as the perturbation to (A + αB) to

get a C0 contraction semigroup with generator(A+ (α+ β)B

).

Iterating this process n times leads to the generator (A + (α + nβ)B),

which is equal to A + B if we choose n such that α + nβ = 1. Such a choice

can always be made with α < 12 and β < 1

2 (1− a) since a < 1. �


Observe that in view of Theorem 3.1.18, the condition that the relatively

bounded operatorB be the generator of a contractionC0-semigroup in Theorem

4.1.3 may be replaced with the condition that B is dissipative and Ran(B−1) =X. However, the range condition may be done away with completely. More

precisely, the following holds (see [19, Corollary 3.3.3]):

Theorem 4.1.5. Let A be the generator of a contraction C0-semigroup {Tt}t≥0

and B be a dissipative operator such that D(B) ⊃ D(A). Suppose B is bounded

relative to A with relative bound less than 1. Then A + B is the generator of

a contraction semigroup.

Remark 4.1.6. We remark here that perturbation of the generator A of a holo-

morphic semigroup by a relatively bounded closed operator B again yields a

holomorphic semigroup (with generator A+B) provided the relative bound is

small enough. We refer the interested reader to [19, Section 3.2], for details of

this and related results.

4.2 Relative boundedness and some consequences

In Definition 4.1.2, the concept of boundedness of an unbounded operator B

relative to another unbounded operator A (in such a case B is said to be A-

bounded, for short) and the bound of B relative to A (often called the A-bound

of B) are defined. Here we collect some properties as consequences of relative

boundedness. For more details, the reader is referred to [1] and [21]. Definition

4.1.2, along with the inequality (4.5), can be rephrased as follows: given two

operators A and B with D(A) ⊂ D(B), B is A-bounded if B maps the set

{x ∈ D(A) : ‖x‖ + ‖Ax‖ ≤ 1} into a bounded set in X. Similarly, B is said to

be A-compact ifD(A) ⊂ D(B) and B maps the set {x ∈ D(A) : ‖x‖+‖Ax‖ ≤ 1}into a pre-compact set or equivalently, for any sequence {xn} ∈ D(A) ⊂ D(B)

such that ‖xn‖+‖Axn‖ ≤ 1, {Bxn} contains a convergent subsequence. Clearly,if B is A-compact, then B is A-bounded. Indeed, let {xn} ∈ D(A) be such that

‖xn‖+‖Axn‖ ≤ 1 and that {‖Bxn‖} is unbounded, making finding a convergent

subsequence in {Bxn} impossible, contradicting the A-compactness of B.

Theorem 4.2.1. Let A : D(A) → X and B : D(B) → X be linear operators

with D(A) ⊂ D(B).

4.2. Relative boundedness and some consequences 87

(i) Suppose that either B is A-bounded with A-bound of B less than 1 or B

is A-compact. If A is closable, so is T = A + B, and D(T ) = D(A). Inparticular, if A is closed, then so is A+B.

(ii) If furthermore, A is the generator of a C0-semigroup in X , then B is A-

bounded if and only if BR(z, A) ∈ B(X), for some z ∈ ρ(A) (and hence

for all z ∈ ρ(A)). Also, in such a case, B is A-compact if and only if

BR(z, A) is compact for some z ∈ ρ(A) (and hence for all z ∈ ρ(A)).

Proof. (i) Note that D(T ) = D(A) and let {xn} ⊂ D(A) be a convergent

sequence such that {Txn} is convergent. Then the inequality (4.5) yields that

‖A(xn − xm)‖ ≤ ‖T (xn − xm)‖+ ‖B(xn − xm)‖≤ ‖T (xn − xm)‖+ a‖A(xn − xm)‖+ b‖xn − xm‖

or, ‖A(xn − xm)‖ ≤ b

(1 − a)‖xn − xm‖+ 1

(1− a)‖T (xn − xm)‖,

since by hypothesis, a < 1, and this implies that {Axn} is convergent. Thus if

xn → 0, and such that Txn → y, then Axn converges. Since A is closable this

means that Axn → 0 and the inequality (4.5) with xn replacing x tells us that

Bxn → 0, so that y = 0. This proves the closability of T. If T and A are the

closures of T and A respectively, then for any x ∈ D(T ), there is a sequence

{xn} ∈ D(A), such that {xn} and {Txn} are both convergent. As we have seen

however, this implies that {Axn} is also convergent, that is, D(T ) ⊂ D(A). The

other inclusion follows similarly (Exercise 4.2.2).

Now, let B be A-compact and let D(A) � xn → 0 and Txn → y. We

need to show that y = 0. We claim that the convergence of {xn} and of {Txn}(which implies the boundedness of both of them) implies that {Axn} contains abounded subsequence. If not, then ‖Axn‖ → ∞ as n→∞. Assume this and set

x′n = xn/‖Axn‖. Then x′

n → 0 and Tx′n =

(‖Axn‖)−1

(Txn) → 0 as n → ∞and ‖Ax′

n‖ = 1. By the A- compactness of B, {Bx′n} contains a convergent

subsequence and replacing x′n by a suitable subsequence we may assume that

Bx′n → w. Then, Ax′

n = Tx′n − Bx′

n → −w. Since x′n → 0, and since A

is closable, we must have w = 0. This contradicts the fact that ‖Ax′n‖ = 1

and therefore, {Axn} contains a bounded subsequence. Thus, choosing that

subsequence and continuing to designate the same by {xn}, we have that xn →0, and Txn → y. This implies that ‖xn‖ + ‖Axn‖ ≤ M. Hence by the A-

compactness of B, {Bxn} has a convergent subsequence. Let Bxn → w, by


renaming the subsequence. Therefore, Axn = Txn − Bxn → y − w and since

A is closable, this leads to the conclusion that y − w = 0. On the other hand,

A-compactness of B implies the A-boundedness of B and hence (4.5) implies

that Bxn → 0, so that y = w = 0.

(ii) Let B be A-bounded. We get from (4.5) that for Re z > β and x ∈ X,

‖BR(z, A)x‖ ≤ a‖AR(z, A)x‖+ b‖R(z, A)x‖

≤ a‖x‖+ (b+ |z|a)‖R(z, A)x‖ ≤ ‖x‖[a+

(b+ a|z|)M(Re z − β)

],

where we have used the estimate in Theorem 2.2.7(f). This leads to the result

that BR(z, A) ∈ B(X).

The converse is simpler and is left as an exercise (Exercise 4.2.3).

Now let BR(z, A) be compact for some z ∈ ρ(A), and let {xn} ⊂ D(A)and {Axn} be bounded sequences. Since ‖(z − A)xn‖ ≤ |z|‖xn‖ + ‖Axn‖, itfollows that {(z−A)xn} is also a bounded sequence and therefore the sequence

{Bxn}n = {BR(z, A)(z −A)xn)}n will have a convergent subsequence. Hence

B is A-compact. The converse follows by retracing the steps of this argument.

That the property of boundedness or compactness of BR(z, A) is independent

of the choice of z ∈ ρ(A) is an easy consequence of the resolvent equation (A.1)

and in left as an exercise (Exercise 4.2.4).

�

When the Banach space X is a Hilbert space, there are more specific

results, as given in the next theorem. But before that, we need to introduce

the concept of essential spectrum of a selfadjoint operator, recalling that the

spectrum of A is defined in Appendix A.1.

Definition 4.2.5. Let A be a selfadjoint operator defined on a Hilbert space H.

The discrete spectrum σd(A) of A is defined as

σd(A) := {λ ∈ σ(A) : λ is an isolated eigenvalue of finite mulitplicity}

and the essential spectrum of A is defined as:

σe(A) := σ(A) \ σd(A).

Thus the essential spectrum of A consists of eigenvalues of infinite multi-

plicity, limit points of eigenvalues of A and the continuous spectrum of A.

4.2. Relative boundedness and some consequences 89

Theorem 4.2.6. Let A be a selfadjoint operator in a Hilbert space H and let B

be a symmetric operator. Then the following hold.

(i) If B is A-bounded with A-bound less than 1, then A + B is selfadjoint

with D(A +B) = D(A) and furthermore, if A is bounded below, then so

is A+B.

(ii) If B is A-compact, then the A-bound of B is 0, and hence A+ B is self-

adjoint. Moreover, in this case σe(A + B) = σe(A), that is, the essential

spectrum of A is invariant under a relatively compact, symmetric pertur-

bation.

Proof. (i) If A is a selfadjoint operator in H , then the whole imaginary axis,

except possibly 0, is in the resolvent set and hence for α ∈ (0,∞), y ∈ H,

‖BR(±iα,A)y‖ ≤ a‖AR(±iα,A)y‖+ b‖R(±iα,A)y‖≤ (

a‖AR(±iα,A)‖+ b‖R(±iα,A)‖)‖y‖ = (a+ bα−1)‖y‖,

where a and b are as in the inequality (4.5). This means that

‖B(R(±iα,A)‖ ≤ (a+ bα−1),

which can be made less than 1 by choosing α large enough, since a < 1. Hence,

for such a choice of α, (I+BR(±iα,A))−1 is in B(H). Therefore, for x ∈ D(A),the identity [

(A+B)± iα]x =

[I +BR(±iα,A)](±iα−A)x, (4.7)

along with the criterion of selfadjointness in Theorem A.1.12 implies that A+B

is selfadjoint on D(A). If A is bounded below, say A ≥ γ, (γ ∈ R) that is, there

exists γ ∈ R such that 〈Ax, x〉 ≥ γ〈x, x〉 for all x ∈ D(A), then choose κ < γ,

so that A − κ > 0. Hence R(κ,A) ∈ B(H) by Theorems A.1.13 and A.1.15,

and the preceding argument can be repeated to show that for any z ∈ C with

Re z = κ < γ,

‖BR(z, A)‖ ≤ a‖AR(z, A)‖+ b‖R(z, A)‖≤ amax{1, |γ|(γ − κ)−1}+ b(γ − κ)−1.

This constant can be made less than 1 by choosing κ appropriately, showing

that such z ∈ ρ(A+B) or that A+B is bounded below.


(ii) As we have observed earlier in the previous theorem, if BR(z0, A) is

compact for some z0 ∈ ρ(A), then BR(z, A) is compact for every z ∈ ρ(A).

Since here A is selfadjoint, it is easily seen from Theorem A.1.15 that

‖(A+ i)R(−in,A)‖ ≤ 1 ∀ n = 1, 2, . . . ,

and that the sequence{(A + i)R(−in,A)} converges strongly to 0 as n → ∞

(Exercise 4.2.7). Since

BR(in,A) = −BR(i, A)[(A+ i)R(−in,A)]∗

it follows that the sequence {BR(in,A)} converges to 0 in operator norm as

n → ∞. (This implication is a consequence of the following statement whose

proof is left as (Exercise 4.2.8): If a sequence {Cn} ⊂ B(X) converges strongly

to C and D is a compact operator, then DC∗n → DC∗ in operator norm as

n→∞.) Moreover, for x ∈ D(A),‖Bx‖ ≤ ‖BR(in,A)‖‖Ax‖+ n‖BR(in,A)‖‖x‖. (4.8)

Since limn→∞ ‖BR(in,A‖ = 0, the coefficient of ‖Ax‖ in (4.8) can be made an

arbitrarily small positive number. From the definition of A-bound of B, this

implies that the A-bound is 0. Thus the selfadjointness of A +B follows from

part (i) of this theorem. The proof of the invariance of the essential spectrum

is omitted and can be found in [1]. �

4.3 Convergence of semigroups

First a kind of master theorem for the convergence of a family of semigroups

is proven, from which further results on the convergence of C0-semigroups in a

Banach space follow.

Theorem 4.3.1. Let {T (λ)(t)}t≥0, λ ∈ (0, 1] be a family of C0-semigroups in

a Banach space X satisfying ‖T (λ)(t)‖ ≤ Meβt for some M > 0 and β ∈R, independent of λ, and let {T (t)}t≥0 be another C0-semigroup, acting on

X0, a closed subspace of X , with ‖T (t)‖ ≤ Meβt. Let these semigroups have

generators A(λ) and A respectively. Then the following are equivalent.

(i) For some t0 ∈ (0,∞) and each x ∈ X0,

limλ→0+

supt∈[0,t0]

{‖T (λ)(t)x− T (t)x‖} = 0.

4.3. Convergence of semigroups 91

(ii) The result (i) holds for every t0 ∈ (0,∞).

(iii) The set S ={z ∈ C : Re z > β

} ⊆ ρ(A(λ)) ∀λ ∈ (0, 1] and S ⊆ ρ(A), and

for some z ∈ S and each x ∈ X0,

limλ−→0+

(z −A(λ)

)−1x = (z −A)

−1x.

(iv) The result (iii) holds for every z ∈ S.

(v) There exists a core D for A in X0 (see Appendix A.1), such that for every

x ∈ D, there exists an x(λ) ∈ D(A(λ)) ⊆ X for each λ ∈ (0, 1] satisfying

x(λ) −→ x and A(λ)x(λ) −→ x strongly as λ −→ 0+.

Proof. That (ii) and (iv) imply (i) and (iii) respectively, is trivial. For the

implication (i) ⇒ (ii), note that for t0 ∈ (0,∞), t ∈ [0, t0] and x ∈ X0,∥∥T (λ)(t+ t0)x− T (t+ t0)x∥∥

≤ ∥∥T (λ)(t)(T (λ)(t0)x− T (t0)x)∥∥+

∥∥(T (λ)(t)− T (t))T (t0)x∥∥

≤Meβt(∥∥T (λ)(t0)x− T (t0)x

∥∥)+∥∥(T (λ)(t)− T (t))T (t0)x

∥∥.Therefore, limλ−→0+ sup0≤t≤2t0

∥∥T (λ)(t)x − T (t)x∥∥ = 0. Proceeding this way

in N steps one has that

limλ−→0+

sup0≤t≤Nt0

‖T (λ)(t)x− T (t)x‖ = 0 ∀x ∈ X0,

and therefore (ii) holds.

(iii)⇒ (v): Let D ⊆ X0 be a core for A, and let x ∈ D. Set y = (z−A)x ∈X0 and, for λ ∈ (0, 1], x(λ) = (z−A(λ))−1y. It is clear that x(λ) ∈ D(A(λ)) ⊆ X

for each λ and that (iii) implies that x(λ) converges as λ −→ 0+, to (z−A)−1y,

which is equal to x by the definition of y. Furthermore,

A(λ)x(λ) = A(λ)(z −A(λ))−1y = −y + zx(λ),

which converges to −y + zx = Ax as λ −→ 0+, proving (v).

(v) ⇒ (iv): Note that, since D is dense in X0 and is a core for A in

X0, (z −A)D = X0 for every z ∈ S. By (v), for every x ∈ D ⊆ X0, there

exists a family {x(λ)} with each x(λ) ∈ D(A(λ)) ⊆ X such that x(λ) −→ x and

A(λ)x(λ) −→ Ax as λ −→ 0+. This implies that (z − A(λ))x(λ) −→ (z − A)x.


Therefore, for all x ∈ D,(z −A(λ))−1(z −A)x − x

= (z −A(λ))−1[(z −A)x − (z −A(λ))x(λ)

]+ x(λ) − x −→ 0 as λ −→ 0+,

since∥∥(z −A(λ))−1

∥∥ ≤ (Re z − β)−1 for every z ∈ S and every λ ∈ (0, 1]. Now

set y = (z − A)x with x ∈ D. Then the above convergence, along with the

earlier observation that (z −A)D = X0, yields the conclusion (iv).

Thus the implications (i) ⇔ (ii) and (iii) ⇒ (v) ⇒ (iv) ⇒ (iii) have been

proven, and to complete the equivalence of all five statements it is sufficient to

prove the equivalence (ii) ⇔ (iv). For (ii) ⇒ (iv), fix z ∈ S and note that, by

Theorem 2.2.7(c),[(z −Aλ)−1x− (z −A)−1x

]=

∫ ∞

0

e−zt(T (λ)(t)x − T (t)x

)dt (4.9)

for x ∈ X0 ⊆ X , and∥∥∥∫ ∞

0

(T (λ)(t)x− T (t)x)e−zt dt∥∥∥ ≤ sup

0≤t≤t0

∥∥T (λ)(t)x− T (t)x∥∥ ∫ t0

0

e−(Re z)t dt

+ 2M‖x‖∫ ∞

t0

e(β−Re z)tdt, (4.10)

where we have used the bounds for the C0-semigroups{T (λ)(t)

}and {T (t)}.

Given ε > 0, choose t0 > 0 sufficiently large such that the second term in the

right hand side of (4.10) is < ε/2 and then for that t0, choose λ > 0 sufficiently

small to make the first term on the right hand side of (4.10) < ε/2 by using

(ii). This observation and (4.9) proves (iv).

(iv) ⇒ (ii): For x ∈ X0, we compute the strong derivative and use the

resolvent equation (A.1) to get

d

ds

{T (λ)(t− s)(z −A(λ))−1T (s)(z −A)−1x

}=(T (λ)(t− s)(−A(λ))(z −A(λ))−1T (s)(z −A)−1x

)+(T (λ)(t− s)(z −A(λ))−1T (s)A(z − A)−1x

)= T (λ)(t− s)

[(z −A)−1 − (z −A(λ))−1

]T (s)x.

Thus we have, by integrating the above between 0 and t, that for x ∈ X0,

(z −A(λ))−1(T (λ)(t)− T (t))(z −A)−1

=

∫ t

0

T (λ)(t− s)[(z −A(λ))−1 − (z −A)−1]T (s)x ds. (4.11)


Using the bound for the semigroup T (λ), one has that

supt∈[0,t0]

∥∥(z −A(λ))−1(

T (λ)(t)− T (t))(z −A)−1x

∥∥≤Me|β|t0

∫ t0

0

∥∥∥[(z −A)−1 − (z −A(λ))−1]T (s)x

∥∥∥ ds. (4.12)

By (iv) and by an application of the Dominated Convergence Theorem,

the right hand side of (4.12) converges to zero as λ −→ 0+. By virtue of the

facts that Ran(z − A)−1 = D(A), which is dense in X0, and that the rest of

the factors in the left hand side of (4.12) are uniformly bounded with respect

to t ∈ [0, t0] and λ ∈ (0, 1], we arrive at

limλ−→0+

supt∈[0,t0]

∥∥(z −A(λ))−1(T (λ)(t)− T (t))y∥∥ = 0, ∀y ∈ X0. (4.13)

Next, note that for x ∈ X0,(T (λ)(t)− T (t)

)(z −A)−1x

= T (λ)(t)[(z −A)−1 − (z −A(λ))−1

]x+

(z −A(λ)

)−1[T (λ)(t)x− T (t)x

]+[(z −A(λ)

)−1 − (z −A)−1]T (t)x. (4.14)

Applying (iv) to the first term in (4.14) and (4.13) to the second term of

(4.14) leads to the conclusion that they converge strongly to zero as λ −→ 0+,

uniformly for t ∈ [0, t0].

It is clear that by (iv) the third term in (4.14) converges to zero for every

t as λ→ 0+. To show that this convergence is uniform with respect to t ∈ [0, t0]

we proceed as follows. For x ∈ X0 and any ε > 0, choose y ∈ D(A) ⊆ X0 such

that∥∥x− y

∥∥ < εe−|β|t0(Re z − β)/8M2 and then note that, using (2.7),∥∥[(z −A(λ))−1 − (z −A)−1

]T (t)x

∥∥≤ 2M2e|β|t0(Re z − β)−1‖x− y‖

+

∫ t0

0

∥∥∥[(z −A(λ))−1 − (z −A)−1

]T (s)Ay

∥∥∥ ds+∥∥∥[(z −A(λ)

)−1 − (z −A)−1]y∥∥∥

which can be made less than ε by choosing λ > 0 sufficiently small, uniformly

in t ∈ [0, t0]. Thus combining the above considerations along with (4.14), we

get that limλ−→0+

supt∈[0,t0]

∥∥(T (λ)(t) − T (t))(z − A)−1x∥∥ = 0, and the property (ii)

follows by the density of D(A) in X0. �


Corollary 4.3.2. Let {T (λ)(t)}t≥0 and {T (t)}t≥0 be two C0-semigroups in a

Banach space X satisfying the bounds for all λ ∈ (0, 1] as in the previous

theorem. Suppose furthermore that D is a core of A, such that A(λ)x −→ Ax

as λ → 0+ ∀x ∈ D. Then T (λ)(t) converges strongly to T (t) as λ −→ 0+,

uniformly for t in any compact subset of [0,∞).

Proof. In the statement (v) of the previous theorem, choose x(λ) = x ∈ X =

X0 ∀λ ∈ (0, 1]. The equivalent statement (ii) gives the result. �

The next theorem is a variation on the above corollary in that we just

need the strong convergence of (z0 −A(λ))−1 to an operator which is injective

with dense range for some z0 with Re z0 > β.

Theorem 4.3.3. Let {T (λ)(t)}t≥0 for λ ∈ (0, 1] be a family of C0-semigroups

with generators A(λ), satisfying∥∥T (λ)(t)

∥∥ ≤ Meβt for some real number β

and M > 0, uniformly in λ. Assume furthermore that for some z0 ∈ C with

Re z0 > β, the family (z0 − A(λ))−1 converges strongly as λ −→ 0+ to a

bounded operator K such that K is injective and the range of K is dense in X .

Then there exists a unique C0-semigroup {T (t)}t≥0 with generator A such that

K = (z0 −A)−1 and T (λ)(t) converges strongly to T (t) as λ −→ 0+, uniformly

for t in compact subsets of [0,∞).

Proof. Without loss of generality, set β = 0. First, we need to extend the strong

convergence to an open set in C. For this let

S ={z ∈ C : |z − z0| < M−1 Re z0

}.

Then by the assumption that∥∥T (λ)(t)

∥∥ ≤M ∀t, one has that

∥∥(z0 −A(λ))−1∥∥ ≤M(Re z0)

−1.

This implies, therefore, that the series∞∑

m=0(z0−z)m

(z0−A(λ)

)−mconverges in

operator norm and defines the operator[I +(z − z0)

(z0−A(λ)

)−1]−1in B(X)

for all z ∈ S. It follows from the relation

z −A(λ) = (z0 −A(λ))[I + (z − z0)(z0 −A(λ))−1

]


that the operator z − A(λ) is injective and that its range equals X, since the

same is true for z0 −A(λ). Hence S ⊆ ρ(A(λ)) and

(z −A(λ)

)−1=(z0 −A(λ)

)−1[I + (z − z0)

(z0 −A(λ)

)−1]−1

=

∞∑m=0

(z0 − z)m(z0 −A(λ)

)−(m+1). (4.15)

Since ∥∥Kx∥∥ = lim

λ→0+

∥∥(z0 −A(λ))−1x∥∥ ≤M(Re z0)

−1∥∥x∥∥,

it follows that(z0−A(λ)

)−(m+1)converges strongly to Km+1 as λ −→ 0+ and

∥∥(z0 −A(λ))−(m+1) −Km+1

∥∥ ≤ 2[M(Re z0)−1](m+1).

An application of the Dominated Convergence Theorem to (4.15) yields that

for z ∈ S,(z −A(λ)

)−1converges strongly as λ −→ 0+ to

∞∑m=0

(z0 − z)mKm+1 := R(z),

the series converging in operator norm. Next note that for any z, z′ ∈ S,

R(z)−R(z′) = (z′ − z)R(z)R(z′) = (z′ − z)R(z′)R(z),

that is, R(z) satisfies the resolvent equation (A.1) since for every λ,(z−A(λ)

)−1

does the same. From this, it follows that

R(z) = R(z′)[I + (z′ − z)R(z)]

showing that RanR(z) ⊆ RanR(z′) and by symmetry between z and z′ one

has that RanR(z) = RanR(z′), that is, the range of R(z) is independent of

z ∈ S. Therefore RanR(z) = RanK, which is dense in X by hypothesis. Now

we define a linear operator A in X as follows. Set its domain D(A) = RanK

and

Ax = z0x−K−1x ∀ x ∈ D(A). (4.16)

Since K is injective and has dense range by hypothesis, A is a densely defined

linear operator. Furthermore, from the definition of A it follows that

(z0 − A)Kx = x, for all x ∈ X and K(z0 −A)x = x for all x ∈ D(A),


and therefore z0 ∈ ρ(A) and K = (z0 − A)−1. This and the definition of R(z)

gives that

S ⊆ ρ(A) and R(z) = (z −A)−1 for all z ∈ S.

SinceM ≥ 1, there is z in S such that Re z > (1−M−1)Re z0 ≥ 0 for which one

gets from (4.15) that (z −A(λ))−1 converges strongly to (z −A)−1 as λ→ 0+.

Thus we conclude that {z : Re z > 0} ⊆ ρ(A). Also, since ‖(z − A(λ))−1‖ is

bounded uniformly in λ for every z with Re z > 0, it follows that (z−A(λ))−m

converges strongly as λ→ 0+ to (z −A)−m for every m ≥ 1 and thus

∥∥(z −A)−m∥∥ ≤ lim inf

λ

∥∥(z −A(λ))−m∥∥ ≤M(Re z)−m.

Therefore, in order to apply Theorem 2.3.1 to show that A is the generator of a

C0-semigroup {T (t)}t≥0 in X, it is left only to show that A is closed. For that

let yn ∈ D(A) −→ y such that Ayn −→ u. By the definition of D(A) and (4.16),

there exists a sequence {xn} ⊂ X such that yn = Kxn for every n and in such

a case AKxn = z0Kxn − xn converges to u as n −→ ∞. This implies that

xn = z0yn−Ayn −→ z0y−u by hypothesis and therefore Kxn −→ K(z0y−u)

because K ∈ B(X). Since yn −→ y, one gets that y = z0Ky −Ku. This shows

that

y ∈ RanK = D(A) and u = z0y −K−1y = Ay.

The conclusion of the theorem follows now by applying the Theorem 4.3.1. �

Further applications of many of the results in this chapter can be seen in

chapters 5 through 7.

Chapter 5

Chernoff’s Theorem and its

applications

In this chapter, a very interesting theorem, due to Chernoff [4], is proven and

some of its applications, viz. the Trotter-Kato Product Formula, the Feynman-

Kac Formula and the Central Limit Theorem are given. A proof of the Mean

Ergodic Theorem is also given at the end as a further application of the concept

of C0-group and Stone’s Theorem.

5.1 Chernoff’s Theorem

Theorem 5.1.1. Let X be a Banach space and let F : [0,∞) −→ B(X) be

a map satisfying ‖F (t)‖ ≤ 1 for all t and F (0) = I. Suppose furthermore

that F ′(0), the strong derivative of F at 0 exists on a dense set D ⊆ X and

that A ≡ F ′(0)|D is the generator of a contraction C0-semigroup {T (t)}t≥0.

Then the sequence{[F(tn

) ]n}nconverges strongly to T (t), uniformly for t in

compact subsets of [0,∞).

We start with a lemma.

Lemma 5.1.2. Let L be a contraction in X . Then for every n ∈ N, and every

x ∈ X , ∥∥[en(L−I) − Ln]x∥∥ ≤ √n

∥∥(L− I)x∥∥.


98 Chernoff’s Theorem and its applications

Proof. Note that for x ∈ X,(en(L−I)−Ln

)x = e−n

∞∑r=0

nr

r!

(Lr−Ln

)x and thus

∥∥(en(L−I) − Ln)x∥∥ ≤ e−n

∞∑r=0

nr

r!

∥∥L|r−n|x− x∥∥

≤ e−n∞∑r=0

nr

r!

∥∥∥ |r−n|−1∑j=0

(Lj+1x− Ljx

)∥∥∥≤(e−n

∞∑r=0

nr

r!|r − n|

)∥∥Lx− x∥∥

≤ √n∥∥(L− I)x

∥∥.The last inequality above is obtained by using the result that(

e−n∞∑r=0

nr

r!|r − n|

)2

≤ e−n∞∑r=0

nr

r!(r − n)2 = n,

which follows by an application of Cauchy’s inequality with respect to the

Poisson probability distribution{

e−nnr

r!

}r=0,1,2,...

. �

Proof of Theorem 5.1.1. Note that for t = 0, F ( tn ) = F (0) = I and T (0) = I

as well and therefore, we can assume without loss of generality that t > 0. Set

An = n[F ( t

n )− I]for a fixed t > 0 and for n = 1, 2, . . . . Then

∥∥An

∥∥ ≤ 2n and

if we set for s ∈ [0,∞), T (n)(s) = exp(sAn), it is clear that {T (n)(s)}s≥0 is a

C0-semigroup, continuous in operator norm. From the contractivity property

of F (t), it follows that∥∥T (n)(s)

∥∥ ≤ 1 for all s and n. Since by hypothesis, Anx

converges strongly to tAx as n −→ ∞ for any x ∈ D, and D is a core (see

Appendix A.1) for the generator tA of a C0 contraction semigroup {T (st)}s≥0

for fixed t > 0, it follows by Corollary 4.3.2 that T (n)(s) converges strongly to

T (st) as n −→ ∞ for the fixed t > 0, uniformly for s in compact subsets of

[0,∞). On the other hand, by setting L = F ( tn ) in Lemma 5.1.2, we get that,

n(L− I

)= An and that for x ∈ D∥∥[T (n)(1)− F

(tn

)n]x∥∥ =

∥∥(en(L−1) − Ln)x∥∥ ≤ √n

∥∥(F ( tn

)− I)x∥∥

≤ tn− 12

∥∥∥nt

(F(tn

)− I)x∥∥∥

≤ t0n− 1

2

∥∥∥nt

(F(tn

)− I)x∥∥, for 0 < t ≤ t0.

Since for x ∈ D, F ′(0)x exists as a strong derivative, it follows that

n

t

(F(tn

)− I)x converges to Ax as n −→∞.

5.1. Chernoff’s Theorem 99

Hence, for all ε > 0, there exists δ > 0 such that∥∥nt

(F(tn

)− I)x−Ax

∥∥ ≤ ε if 0 < t/n < δ

whereas ∥∥nt

(F(tn

)− I)x∥∥ ≤ 2n

t ‖x‖ ≤ 2δ−1‖x‖ if t/n ≥ δ.

Thus,

supn

supt∈[δ,t0]

∥∥nt

(F( tn

)− I)x∥∥ ≤ max{ε+ ‖Ax‖, 2δ−1‖x‖}.

This, combined with the earlier observation that T (n)(1) converges to T (t),

implies that for any x ∈ D, [F ( tn )]

nx converges strongly to T (t)x as n −→∞,

uniformly for t ∈ [0, t0]. Finally, the density of D in X and the facts that

both F (t) and T (t) are contractions allow the extension of this convergence,

uniformly for t ∈ [0, t0], to the whole space X . �

The proof of the following result uses Chernoff’s Theorem 5.1.1.

Theorem 5.1.3 (Trotter-Kato Product). Let {T (t)}t≥0 and {S(t)}t≥0 be two

contractive C0-semigroups with generators A and B respectively. Assume fur-

thermore that D(A) ∩ D(B) ≡ D is dense in X and that (A +B)|D is the

generator of a C0-semigroup {Z(t)}t≥0. Then the sequence (T (t/n)S(t/n))n

converges strongly to Z(t) for each t ≥ 0 and the convergence is uniform for t

in any compact subset of [0,∞).

Proof. Define F : [0,∞) → { contractions in X} by F (t) = T (t)S(t) for all

t ≥ 0. Then F (0) = I and for any x ∈ D,∥∥∥(F (t)− I

t

)x− (A+B)x

∥∥∥≤∥∥∥T (t)(S(t)− I

t−B

)x∥∥∥+ ‖(T (t)− I)Bx‖ +

∥∥∥(T (t)− I

t−A

)x∥∥∥

−→ 0 as t −→ 0,

that is, F ′(0)x exists for every x ∈ D and equals (A + B)x. By hypothe-

sis, (A+B)|D is the generator of a C0-semigroup {Z(t)}t≥0 and therefore by

Chernoff’s Theorem,

s− limn−→∞

(T (t/n)S(t/n))n = s− limn−→∞

F (t/n)n = Z(t),

uniformly for t in compact subsets of [0,∞). �


Remark 5.1.4. 1. Theorem 5.1.3 subsumes the classical Lie Product Formula

which motivated the above generalisation. The Lie Product theorem is as

follows: let A and B be two complex n× n matrices. Then (etA/netA/n)n

converges to et(A+B) as n → ∞, uniformly for t in compact subsets of

[0,∞).

Note that even if the semigroups concerned are not necessarily con-

tractive, but their generators are bounded operators in X, we can work

with their contractive modifications, viz. et(A−‖A‖) and et(B−‖B‖) respec-

tively, to get the limiting semigroup to be et(A+B−(‖A‖+‖B‖)), and thereby

to the same result after an application of Theorem 5.1.3.

2. There are several further generalisations of Theorem 5.1.3. We state here

only one of them without proof and refer the reader to the original source

[14]. For definitions and explanations of the terms used in Theorem 5.1.5

and a sketch of the proof, the reader is referred to [21] and the article of

[14].

Theorem 5.1.5. Let {T (t)}t≥0 and {S(t)}t≥0 be two holomorphic semigroups

with generators A and B respectively, in a Hilbert space H, so that −Aand −B are m-sectorial, with associated forms a and b. Assume furthermore

that D(a) ∩ D(b) is dense in H and such that the associated sectorial form

a + b is closed. Then the Trotter-Kato Product Formula is valid, that is,

s− limn−→∞

(T (t/n)S(t/n))n = Z(t), uniformly for t in compact subset of [0,∞),

where {Z(t)}t≥0 is the holomorphic semigroup generated by C, with −C the

m-sectorial operator associated with the form a+ b.

5.2 Applications of the Trotter-Kato Product Formula and of

Chernoff’s Theorem

Example 5.2.1. let H = L2(R) and let V be a real-valued measurable function

with V = V1+V2 where V1 ∈ L2(R) and V2 is bounded. Then MV , the operator

of multiplication by the function V, is selfadjoint (see Theorem A.1.13) and we

set (T (t)f)(x) = eitV (x)f(x) for f ∈ H to define {T (t)}t∈R as a unitary C0-

group with generator A = iMV . Let {S(t)}t∈R be the translation group in Hgiven by (S(t)f)(x) = f(x+ t) for all f ∈ H and x, t ∈ R. The generator B of

5.2. Applications of Trotter-Kato and Chernoff Theorem 101

the group {S(t)}t∈R is easily seen to be B = iP, where P =(− i d

dx

)|S(R) and

S(R) is the class of smooth functions of rapid decrease (see Appendix A.2).

Next we note that A+B = i(P +MV ) and we claim that P +MV is selfadjoint

on D(P ).

This is because if f ∈ D(P ) ⊆ L2(R), then f is bounded, uniformly

continuous and converging to 0 as |x| −→ ∞ (see Remark A.2.6(d)) and it

follows that D(P ) ⊂ D(MV ). Furthermore, for x ∈ D(P ),

‖MV x‖ ≤ ‖MV1(P + in)−1‖‖Px‖+ (‖V2‖∞ + n‖MV1(P + in)−1‖)‖x‖

and since

‖MV1(P + in)−1‖ ≤ (2π)−1/2‖V1‖2( ∫

R

|k + in|−2 dk)1/2

≤ (2n)−1/2‖V1)‖2 → 0 as n→∞,

it follows that MV has P -bound 0 (see Definition 4.1.2). Thus, by Theorem

4.2.6, P +MV is selfadjoint and we apply the Trotter-Kato Product Formula

to get that

Z(t) ≡ s− limn−→∞

(T (t/n) S(t/n))n,

where {Z(t)}t∈R is the unitary group generated by i(P +MV ). A simple com-

putation yields that for any f ∈ L2(R) and t ∈ R,

((Tt/nSt/n

)nf)(x) = exp

(in−1∑j=0

( t

n

)V(x+

jt

n

))f(x+ t), (5.1)

for almost all x ∈ R. We claim that for our choice of the function V, the right

hand side of (5.1) converges as n→∞ to exp(i

∫ t

0

V (x + s) ds)f(x+ t). For

this, we note that for V in the class mentioned above one can find a sequence

{Vn} of real finitely valued functions approximating V in the L1loc(R) topology

and therefore it suffices to show that for a bounded interval � ⊆ R,

∫dx∣∣∣ n−1∑j=0

∫ t(j+1)/n

tj/n

[χ�

(x+ s

)− χ�(x+ tj/n

)]ds∣∣∣ −→ 0 as n −→∞.

Indeed this follows from the fact that for s ∈ (tj/n, t(j + 1)/n),∫

dx∣∣χ�

(x+ s

)− χ�(x+ tj/n

)∣∣ = 2(s− tj/n

)


and this in its turn implies that exp(i

n∑j=1

(t/n

)V(x + jt/n

))converges to

exp(i

∫ t

0

V(x + s

)ds)pointwise almost everywhere in x, by choosing a sub-

sequence if necessary. Finally, an application of the Dominated Convergence

Theorem to (5.1) shows that(T (t/n)S(t/n)

)nconverges strongly to the map

Z(t) as n −→∞, where Z(t)f in L2(R) is given as

(Z(t)f

)(x) = exp

(i

∫ t

0

V(x+ s

)ds)f(x+ t). (5.2)

However, this could also have been obtained simply by observing that if we set

(Wf

)(x) = exp

(i

∫ x

0

V(y)dy)f(x),

then (i) W defines an unitary operator in L2(R) which leaves D(P ) invariant

and (ii) it intertwines between the selfadjoint operators P and P +MV , that is,

PWf = W (P +MV )f for all f ∈ D(P ). Then Z(t) = W ∗S(t)W and a simple

computation of the right hand side of this equality on a vector f ∈ L2(R)

establishes (5.2).

Example 5.2.2. (Feynman-Kac Formula) Let {T (t)}t≥0 be the heat semigroup

in L2(Rd), as described in Example 2.5.5 (see Theorem 2.5.9 in particular). Its

generator is H0 ≡ 12Δ, the Laplacian in Rd, with its domain of selfadjointness

D(H0) ={f ∈ L2(Rd) :

∫|k|4|f(k)|2dk <∞

},

where f is the L2-Fourier transform of f (see Appendix A.2). Further, let

{S(t)}t≥0 be the C0-semigroup given by(S(t)f

)(x) = e−tV (x)f(x) for all f ∈

L2(Rd), t ∈ R and x ∈ Rd, where V ∈ L∞(Rd) is fixed (see Example 2.5.4).

We note that though {T (t)}t≥0 is a contraction semigroup, {S(t)}t≥0 is not.

However, that can be easily taken care of by setting

(S(t)f)(x) = e−t(V (x)+‖V ‖∞

)f(x) = e−t‖V ‖∞(S(t)f)(x),

so that the new operator S(t) differs from S(t) by multiplication by a scalar

and {S(t)}t≥0 is now a contraction C0-semigroup. This does not make material

change in what is going to follow. In fact, now we can apply Theorem 5.1.3 to

get that Z(t) = s− limn−→∞

(T (t/n)S(t/n)

)n, where Z(t) is a C0-semigroup with


generator (H0 + V ), which is a selfadjoint operator with domain D(H0). We

know the kernel of T (t) (see Example 2.5.5): if f ∈ L2(Rd), then

(T (t)f

)(x) =

∫Rd

K(t;x− y)f(y) dy, with

K(t;x) = (2πt)−d/2 exp(− |x|

2

2t

)for t > 0. (5.3)

One can explicitly compute to get

((Tt/nSt/n

)nf)(x)

=

∫. . .

∫K(t/n;x1 − x0)e

−t/nV (x1) . . .K(t/n;xn − xn−1)×

e−t/nV (xn)f(xn) dx1 . . . dxn

= (2πt/n)−nd/2

∫. . .

∫exp

(− 1

2

n∑j=1

(t/n)(xj − xj−1

t/n

)2)×exp

(−

n∑j=1

(t/n)V (xj))f(xn) dx1 . . . dxn, (5.4)

where we have set x0 = x. Since f ∈ L2(Rd), strong convergence implies

convergence pointwise almost everywhere for a subsequence, one expects that

for almost all x ∈ Rd,(e−t(H0+V )f

)(x) will equal the limit of right hand side

of (5.4), as n → ∞ by choosing to enumerate the subsequence appropriately.

Here we face a few difficulties in taking the limit:

•n∏

j=1

dxj , the product of n Lebesgue measures in Rd has no reasonable limit

as a measure as n −→∞;

• the factor (2πt/n)−nd/2 also does not have any limit;

• if we think of the points xj−1, xj (j = 1, 2, . . .) as the initial and fi-

nal points respectively of a piecewise linear path w : [0, t] −→ Rd, then

the expression in the second exponential in (5.4) formally converges to

−∫ t

0

V (w(s))ds, while that in the first exponential formally converges to

− 12

∫ t

0

[dwds

(s)]2ds.

Next, we note that for most (continuous) paths w, w will be quite singular

leading to the formal vanishing of the first exponential factor in (5.4) while the


n∏j=1

dxj diverges as n −→ ∞, leaving us with the tantalising possibility that

the product somehow will be meaningful. This we address next by constructing

the Wiener measure P on the linear space of continuous paths w, making sense

of the limit as n→∞ of the right hand side of (5.4) as∫e−

∫ t0V (x+w(s)) dsf(x+ w(t))P(dw),

where the integration is over the (infinite dimensional) linear vector space

C(R+,Rd) of all Rd-valued continuous paths w.

There are a few different ways of constructing the Wiener measure. We

shall present here in brief one such method, more probabilistic in spirit, using

Kolmogorov’s Extension or Consistency Theorem [17, pages 143–144 and 212–

224]. For an alternative approach to the construction of the Wiener measure,

the reader is referred to [21, pages 277–279].

The expressions (2.29) and the associated heat kernel K in (5.3) and the

right hand side of (5.4) admits probabilistic interpretations as follows. For every

Borel set B in Rd × . . .× Rd︸︷︷︸n-fold

define the family of distributions Ft1,t2,t3,...,tn(B)

for 0 < t1 < t2 < · · · < tn < t < τ <∞ as

Ft1,t2,t3,...,tn(B)

=

∫B

p(0, x; t1, x1)p(t1, x1; t2, x2) · · · p(tn, xn; t, y) dx1 dx2 · · · dxn, (5.5)

where p(s, x; t, y) = K(t − s; y − x). The quantity Ft1,t2,t3,...,tn(B) is the n-

dimensional probability distribution that the stochastic process, starting at

time 0 at x ∈ Rd will arrive at time t at y ∈ Rd, after having passed through

the region B in Rdn at time-sequence t1 < t2 < . . . < tn. The collection{Ft1,t2,...,tn(B) : 0 < t1 < t2 < . . . < tn < t < τ ;B a Borel set in (Rd)n;n ∈ N

}is called the set of all finite-dimensional distributions of the stochastic process.

It is clear that

(i) Ft1,t2,...,tn(B) is increasing if B is increasing;

(ii) Ft1,t2,,...,tn(·) is countably additive;

(iii) Ft1,t2,...,tn(Rdn) = p(0, x; t, y), which follows from the properties of a con-

volution semigroup (see Exercise 2.5.8 and Theorem 2.5.7), and


(iv) The distributions Ft1,t2,...,tm are consistent. This means that for all Borel

sets E1, E2, . . . Em in Rd,

Ft1,t2,...,tm(E1 × . . .× Em)

= Fs1,s2,...,sn

(Rd × . . .× Rd︸︷︷︸

n0

×E1 × Rd × . . .× Rd︸︷︷︸n1

×E2

× . . .× Em × Rd × . . .× Rd︸︷︷︸nm

),

where {t1, t2, . . . , tm} ⊂ {s1, s2, . . . , sn} ⊂ [0, τ ],m < n such that

s(0)1 < . . . < s(0)n0

< t1 < s(1)1 < . . . < s(1)n1

< t2 < . . . < tm < s(m)1 < . . . < s(m)

nm.

The notation above means that there are n0 sj ’s before t1, n1 sj ’s between

t1 and t2 and so on. As was the case for (iii), the consistency property (iv)

of the distributions F can be easily verified for the distribution generated

by the heat kernel K. (Exercise 5.2.3).

Then by Kolmogorov’s Extension Theorem [17, Chap 5, Theorem 5.1] there

exists a probability space (Ω,�,Px) with a unique measure Px and a stochastic

process

X : R+ × Ω→ Rd which is �-measurable and such that

P(w ∈ Ω : (X(0, w) = x,X(t1, w), . . . , X(tn, w)) ∈ B) = Ft1,t2,...,tn+1(B)

for all B ∈ Borel subsets of (Rd)n. This stochastic process {X(t)}t≥0 is called

the Brownian motion, starting at x ∈ Rd, and the associated measure Px the

Wiener measure . If we set x = 0 in (5.5), that is, if the process starts at the

origin, then X(0) = 0 and in such a case {X(t)}t≥0 is the standard Brownian

motion (S.B.M.)

Further properties of S.B.M. are as follows.

1. X(0) = 0,

2. Xj(t)−Xj(s) is distributed as a standard normal distribution N(0, t− s)

for 1 ≤ j ≤ d and 0 < s < t. This can be seen by computing the variance∫R

(xj − yj)2p(s, xj ; t, yj) dyj = [2π(t− s)]−1/2

(∫R

x2je

− x2j

2(t−s) dxj

)= t− s.


3. {X(t)} is an independent increment process, that is,

X(t1), X(t2)−X(t1), X(t3)−X(t2), . . . , X(tn)−X(tn−1)

are independent Rd-valued random variables for all 0 ≤ t1 < t2 < t3 . . . <

tn−1 < tn.

4. The Kolmogorov Continuity Criterion [17, Chap 7, Theorem 3.1] states

that if there are constants α, δ, κ > 0 such that

E(|X(t)−X(s)|α) ≡ ∫

|u− v|α dFt,s(u, v)

≤ K(t− s)(1+δ) for all 0 < t < s < τ,

then there exists a unique measure μ on C[0, τ ] such that all the finite-

dimensional distributions associated with μ equal Ft1,t2,...,tn(·) from which

the measure Px was constructed. In the above expression E is the expec-

tation. It is easy to see (Exercise 5.2.4) that if X(·) are the Rd-valued

S.B.M’s, then for p ≥ 2, 0 ≤ s < t, we have

E( d∑

j=1

|Xj(t)−Xj(s)|2)p/2

= (t− s)p/2(E|ξ|p),

where ξ is an Rd-N(0, 1) random variable. This implies that the Wiener

measure P, constructed to support the stochastic process which is the

S.B.M. is actually equivalent to a probability measure on C[0, τ ], and we

shall identify the two and call it also the Wiener measure.

In some sense the original probability space is not relevant any more and

we can assume that the S.B.M. (or the Rd-valued Wiener process) is the process

R+ × C(R+) �→ Rd given by (t, w) �→ w(t), that is, work with the continuous

paths on R+ and the Wiener measure on them.

Now we are in a position to re-interpret (or rewrite) the expression (5.4)

for((T (t/n)S(t/n)

)nf)(x) as

=

∫Ω

exp{−

n∑j=1

(t/n)V (w(tj/n))}f(w(t))Px(dw)

=

∫Ω

exp{−

n∑j=1

(t/n)V (x+ w(tj/n))}f(x+ w(t))P(dw), (5.6)


where we have written Ω for C(R+). As in the case of Example 5.2.1, one can

approximate V by a sequence of uniformly bounded simple functions, pointwise

and since for a bounded Borel set � ⊆ Rd,

n∑j=1

∫P(dw)

∫Rd

dx

∫ tj/n

t(j−1)/n

ds∣∣(χ�(x+ w(s)) − χ�(x+ w(tj/n))

∣∣≤

n∑j=1

∫ tj/n

t(j−1)/n

ds

∫P(dw)

d∏k=1

∣∣2[wk(s)− wk(tj/n)]∣∣. (5.7)

By the Cauchy-Schwartz inequality, the expression on the right hand side of

(5.7) is

≤ 2dn∑

j=1

∫ tj/n

t(j−1)/n

ds[ ∫

P(dw)d∏

k=1

∣∣wk(s)− wk(tj/n)∣∣2]1/2

= 2dn∑

j=1

∫ tj/n

t(j−1)/n

|s− tj/n|d/2 ds −→ 0 as n −→∞. (5.8)

To arrive at (5.8), we have used the property that for 0 ≤ s < t,

∫P(dw)

d∏k=1

∣∣wk(s)− wk(t)∣∣2 =

[ ∫K(t− s;x)x2 dx

]d= (t− s)d.

Thus it follows that for V ∈ L∞(Rd),∫�

∣∣∣ exp{− n∑j=1

(t/n)V (x + w(tj/n))}− exp

{−∫ t

0

V (x+ w(s))ds}∣∣∣ dx

converges to 0 as n −→ ∞ for every bounded set � and hence the function

inside | · | converges to 0 pointwise for almost all x ∈ Rd and for almost all w

(if necessary, by choosing a subsequence). Finally, since

∣∣∣ exp{− n∑j=1

(t/n)V (x+ w(tj/n))}∣∣∣ ≤ et‖V ‖∞ for all x and w,

an application of the Dominated Convergence Theorem to the right hand side

of (5.6) leads to the conclusion that

((T (t/n)S(t/n))nf)(x) −→∫Ω

exp{− ∫ t

0

V (x+ w(s))ds}f(x+ w(t))P(dw)

=

∫Ω

exp{− ∫ t

0

V (w(s))ds}f(w(t))Px(dw), (5.9)


for almost all x ∈ Rd as n −→∞. Thus,

(Z(t)f)(x) =(e−t(H0+V )f

)(x) =

∫Ω

exp{−∫ t

0

V (w(s))ds}f(w(t))Px(dw)

(5.10)

which is the Feynman-Kac formula.

Remark 5.2.5. Feynman’s original idea ([10]) was to look for a description of

the solution of the Schrodinger equation (which in quantum mechanics gov-

erns the evolution of the wave function, describing the state of the associated

quantum mechanical physical system) in terms of all possible continuous paths

connecting the initial space time point with the final one (see also Section 7.3).

Unlike in the above discussion on the Feynman-Kac Formula, the relevant ve-

hicle there is the unitary group given by U(t) = exp(− i(H0+V )t

), and if one

tries to write down an expression similar to that in (5.4) it would be

(U(t)f)(x) = limn−→∞

(2πitn

)−nd/2∫

...

∫exp

{− i

2

n∑j=1

(t/n)(xj − xj−1

t/n

)2}×exp

{− in∑

j=1

t/nV (xj)}f(xn)dx1...dxn, (5.11)

that is, it replaces t in (5.4). However, unlike in the earlier case, the formal

limit of the expression in the right hand side of (5.11) as n → ∞ cannot be

made sense of. That is, there does not exist any measure νx on Cx(R+), the

space of continuous functions f on R+ with f(0) = x, such that (5.11) may be

reinterpreted as

∫Cx(R+)

exp{− i

∫ t

0

V (w(s)) ds}f(w(t))νx(dw),

as was possible in the earlier case of the construction of the Wiener measure.

Example 5.2.6 (Central Limit Theorem). Let X and Y be two independent real-

valued random variables with distributions G and H respectively; it is known

(see [18]) that X +Y is a real-valued random variable with distribution G ∗H .

Explicitly, let

Probability {X ≤ x} ≡ Pr{X ≤ x} = G(x) and Pr{Y ≤ y} = H(y),


and then

Pr{X + Y ≤ α} = Pr{X ≤ x and Y ≤ y | x+ y ≤ α}

=

∫R

G(α− y)H(dy) ≡ (G ∗H)(α) = (H ∗G)(α).

Next, let {ξi}ni=1 be n independent real-valued random variables, each with

distribution G, independent of i and with mean 0 and variance 1. Then by a

simple extension of the above line of reasoning, one has that for α ∈ R,

P r{ 1√

n

n∑i=1

ξi ≤ α}= (Gn ∗Gn ∗ ... ∗Gn)︸︷︷︸

n-fold

(α),

where Gn(x) = G(√nx). It is natural to set this problem up in some Banach

space X, say in X ≡ C0(R), the Banach space of continuous complex-valued

functions vanishing at infinity, equipped with the supremum norm, and to asso-

ciate with the action of convolution a linear operator on X . For this, we define

for f ∈ X and any fixed distribution function G on R, the linear operator AG

by

(AGf)(x) =

∫f(x− y)G(dy). (5.12)

Then, one can verify that

(a) AG is a bounded linear operator in X , in fact∥∥AG

∥∥ ≤ 1;

(b) AG1∗G2 = AG1AG2 = AG2AG1 = AG2∗G1 .

For t > 0 and n ∈ N, define F (t/n) = AG(n/t), where G(n/t)(x) = G(

√n/t x) =

Pr{ξ1 ≤

√n/t x

}. Then note that by the observation (b) above,

F (t/n)n =(AG(n/t)

)n= AG(n/t)∗G(n/t)∗...∗G(n/t)

= APr{√

t/n∑

ni=1 ξi≤·

},thereby setting the stage for an application of Theorem 5.1.1. For that we need

first to compute F ′(0), which is done in the next lemma.

Lemma 5.2.7. Let F (t) = AG(1/t)for all t > 0, and define an operator Δ in X

by setting

D(Δ) ={f ∈ X : f ′ and f ′′ ∈ X

}and (5.13)

(Δf)(x) =1

2f ′′(x) for any f ∈ D(Δ) and x ∈ R. (5.14)


Then F ′(0)f = Δf for f ∈ D(Δ), that is,{F (t)− I

tf − Δf

}−→ 0 as t −→ 0+.

Proof. For t > 0 and f ∈ D(Δ),

t−1[F (t)f − f − tΔf

](x) (5.15)

= t−1[AG(1/t)f − f − tΔf ](x)

= t−1

∫R

[f(x− y)− f(x) + yf ′(x) − y2

2f ′′(x)

]G(1/t) (dy)

= t−1

∫R

[f(x−√tu)− f(x)−√tuf ′(x)− tu2/2f ′′(x)

]G(dy) (5.16)

where we have changed variable y =√tu and used the properties that∫

yG(1/t)(dy) =√t

∫yG(dy) =

√tE(ξ1) = 0

and ∫y2G(1/t)(dy) = t

∫u2G(du) = tE(ξ21) = t.

Using Taylor’s Theorem, there exists θ ∈ (0, 1) such that the right hand side

of (5.16)

=1

2

∫R

[f ′′(x − θ

√tu)− f ′′(x)

]u2G(du)

=1

2

∫|u√t|<δ

[f ′′(x − θ√tu)− f ′′(x)]u2G(du)

+1

2

∫|u√t|≥δ

[f ′′(x− θ√tu)− f ′′(x)]u2G(du)

= I1(t) + I2(t).

Since f ′′ ∈ X , it follows that f ′′ is uniformly continuous on R. Therefore given

ε > 0, there exists δ = δ(ε) > 0 such that supx∈R

∣∣f ′′(x − θ√tu) − f ′′(x)

∣∣ < ε

if∣∣θ√tu

∣∣ < |√tu| < δ and therefore∣∣I1(t)∣∣ < ε/2

∫R

u2G(du) = ε/2.

Having chosen this δ, now one chooses t0 with 0 < t0 < 1 such that∣∣I2(t)∣∣ ≤ ∥∥f ′′∥∥∞

∫|u|≥δt−1/2

u2G(du) < ε/2 for t < t0.

This completes the proof.

�


Theorem 5.2.8 (Central Limit Theorem). Let {ξi}i≥1 be a sequence of inde-

pendent and identically distributed real-valued random variables with mean

0 and variance 1. Then as n −→ ∞, Pr{

1√n

n∑i=1

ξi ≤ α}

converges to

1√2π

∫ α

−∞e−x2/2dx, that is, the random variable 1√

n

n∑i=1

ξi converges to a

N(0, 1)-random variable in distribution.

Proof. As we have already seen, if F (t/n) = AG(n/t)where G is the distribution

of each ξi, then F ′(0)f = Δf for f ∈ D(Δ), a dense subset of X . Furthermore,

we have seen in Chapter 2, Example 2.5.5 that Δ is the generator of the heat

semigroup T (t), a contraction C0-semigroup. Therefore one can apply Cher-

noff’s Theorem 5.1.1 to get that

F (t/n)n = AG(n/t)∗···∗G(n/t)= A

Pr{√

t/n∑n

i=1 ξi≤·}

converges to the heat semigroup {T (t)}t≥0, or equivalently, for any f ∈ X ,

(F (t/n)nf

)(x) = Ef

(x−

√t/n

n∑i=1

ξi

)−→ 1√

2πt

∫f(x− y)e−y2/2t dy

as n→∞. �

Example 5.2.9 (Mean Ergodic Theorem). Let (Ω,�, μ) be a finite measure space

and let ξ be a measure preserving transformation on Ω, that is, μ(ξ−1(�)) =

μ(�), ∀� ∈ �. The transformation ξ is said to be ergodic if the only mea-

surable sets (up to a set of measure 0) that are invariant under ξ are the

whole set Ω and the null set ∅. This action can be lifted as a map on the

linear spaces of complex-valued measurable functions on (Ω,�, μ) by setting

(Tξf)(x) = f(ξ(x)). In particular, if we start with f ∈ L2(Ω,�, μ) ≡ X , then

we observe, that

∥∥Tξf∥∥2 =

∫Ω

|f(ξ(x))|2μ(dx) =∫ξ(Ω)

|f(x)|2μ · ξ−1(dx) ≤ ∥∥f∥∥2,that is, Tξ is a contraction on X . If furthermore the map ξ is surjective,

that is, ξ(Ω) = Ω, then the map Tξ is an isometry in X . Finally, if Ω is

a locally compact topological space and � is the Borel σ-algebra of Ω, and

if we have a one-parameter continuous group {ξt}t∈R of measure-preserving

transformations on (Ω,�, μ) (that is, ξ−1t = ξ−t, ξ0 = id, ξtξs = ξt+s for all


t, s ∈ R and ξt(x) −→ x ∈ Ω as t −→ 0), then the family {U(t)}t∈R of

operators in X given by (U(t)f)(x) = f(ξt(x)) ∀ f ∈ X and x ∈ Ω, is a one-

parameter strongly continuous group of unitaries in X . The strong continuity

at t = 0 follows since∫|((U(t)− I)f)(x)|2μ(dx) =

∫|f(ξt(x)) − f(x)|2μ(dx) −→ 0 as t −→ 0

by an application of the Dominated Convergence Theorem with simple func-

tions f. Since the subspace of simple functions is dense in L2(Ω, μ), the above

convergence may be extended to all of L2(Ω, μ). By Stone’s theorem (Theorem

3.2), one has iA as the generator of U(t), for some selfadjoint operator A in X .

Next, the Mean Ergodic Theorem is stated and proven.

Theorem 5.2.10 (Mean Ergodic Theorem). Let {ξt}t∈R be a one-parameter

continuous group of measure-preserving, ergodic transformations on (Ω,�, μ)

and let X = L2(Ω, μ). Also, let {U(t)}t∈R be the associated strongly continuous

unitary group in X with iA as its generator. Then for every f ∈ X , the time

average of f under the evolution {U(t)}t∈R, the strong limit

limT→∞

1

2T

∫ T

−T

U(s)f ds

exists and equals Pf, where P is the projection onto N (A), the null space of

A. Furthermore, the subspace P (X) consists of functions which are constant μ

-almost everywhere and (Pf)(x) = [μ(Ω)]−1[ ∫

f(y)μ(dy)](the space average)

∀ f ∈ X .

Proof. First of all, let us make the simple observation that

f ∈ N (A) ≡ {f ∈ D(A) ⊆ X : Af = 0

}if and only if U(t)f = f for all t ∈ R.

If f ∈ N (A) or equivalently if U(t)f = f, then the group property implies

that given any x ∈ Ω there exists y ∈ Ω such that ξty = x and hence f(y) =

(U(t)f)(y) = f(ξt(y)) = f(x). This means that such an f is constant μ-almost

everywhere and also that for any g ∈ X,∫(Pg)(x)μ(dx) = (Pg)(x)μ(Ω) = 〈Pg, 1〉 = 〈g, P1〉 = 〈g, 1〉 =

∫g(x)μ(dx),


since (U(t)1)(x) = 1( that is 1(ξt(x)) = 1 ∀t, for almost all x ∈ Ω). It is easy

to see that if g ≡ limT→∞ 12T

∫ T

−T U(s)fds exists, then U(t)g = g. In fact,

U(t)g = limT→∞

1

2T

∫ T

−T

U(t+ s)fds = limT→∞

1

2T

∫ T+t

−T+t

U(s)f ds

= limT→∞

1

2T

∫ T

−T

U(s)f ds+ limT→∞

1

2T

∫ T+t

T

U(s)f ds

+ limT→∞

1

2T

∫ −T

−T+t

U(s)fds = g,

since the second and the third limits vanish. This leads to the conclusion that

g ∈ N (A), and furthermore,

g = Pg = limT→∞

1

2T

∫ T

−T

PU(s)f ds = limT→∞

1

2T

∫ T

−T

Pf ds = Pf.

For the penultimate equality, note that U(t)P = P for all t ∈ R so that

PU(t) = P for all t ∈ R on taking adjoints. Hence, for every f ∈ X, if

limT→∞

1

2T

∫ T

−T

U(s)f ds ≡ g exists, then g = Pf, and by the earlier discussions,

g(x) = (Pf)(x) = [μ(Ω)]−1[ ∫

f(y)μ(dy)].

Thus to complete the chain of reasoning, we have to prove the existence of the

above-mentioned limit g. For this, note first that X = N (A) ⊕ Ran(A) since

A is selfadjoint. For arbitraryf ∈ X, set f = Pf + f , with f ∈ Ran(A), and

note that1

2T

∫ T

−T

U(s)Pf ds = Pf. On the other hand, if h ∈ Ran(A), that is,

h = Au for some u ∈ D(A), then1

2T

∫ T

−T

U(s)h ds =1

2T

∫ T

−T

U(s)Auds = − i

2T

∫ T

−T

d

dsU(s)u ds

= −i[U(T )u− U(−T )u

2T

]−→ 0 as T →∞.

For f ∈ Ran(A) and any ε > 0, one can find h ∈ Ran(A), such that ‖f−h‖ < ε,

and one has that

limT→∞

1

2T

∫ T

−T

U(s)f ds = limT→∞

1

2T

∫ T

−T

U(s)(f − h) ds

+ limT→∞

1

2T

∫ T

−T

U(s)h ds

= limT→∞

1

2T

∫ T

−T

U(s)(f − h) ds,


which implies that

∥∥ limT→∞

1

2T

∫ T

−T

U(s)f∥∥ ≤ ∥∥ lim

T→∞1

2T

∫ T

−T

U(s)(f − h) ds∥∥ < ε.

Since ε > 0 is arbitrary, we conclude that

limT→∞

1

2T

∫ T

−T

U(s)f = 0.

�

Chapter 6

Markov semigroups

In Chapter 4, we have seen methods of constructing a semigroup by perturbing

a known (or given) one, where the perturbation is small in a certain sense.

More precisely, the context were those in which the candidate for the generator

of the new semigroup was obtained by a small additive perturbation of the

generator of the known semigroup. On the other hand, in many applications,

particularly in the theory of probability and stochastic processes, situations

arise when the perturbation is comparable (not small) to the generator of the

known semigroup and in such cases, the theory studied in Chapter 4 is not

applicable. However, often this difficulty can be circumvented by exploiting

the property of positivity or positivity-preserving of the semigroup (and of the

resolvents of the generator involved.) We shall concern ourselves with such

situations in this chapter, mostly restricting ourselves to a concrete model of

the construction of a Markov semigroup, following Kato [13].

6.1 Probability and Markov semigroups

Any theory of classical probability starts with a state space(Ω,�, μ

)where Ω

is a measure space, � the σ-algebra of measurable subsets (events) of Ω and

μ is a probability measure on �, that is, μ : �→ [0, 1] is a countably additive

map such that

0 ≤ μ(F ) ≤ 1 ∀F ∈ �, μ(∅) = 0 and μ(Ω) = 1.


116 Markov semigroups

The random variables are measurable maps or functions (real, complex or more

generally vector-space valued) f : Ω → X, and their X- valued expectation is

given by

E(f) =

∫f(w)μ(dw).

In fact, the linear space of such functions (called random variables) and the

associated probability measure are in dual relationship. More precisely, for ex-

ample, if X = C and Ω = [0, 1], equipped with the standard Borel structure,

and the space of random variables on Ω is C[0, 1], then the Riesz-Markov Theo-

rem [24, Theorem 6.19], tells us that every normalised positive linear functional

φ on C[0, 1] is given by a regular probability measure μφ such that

φ(f) =

∫[0,1]

f(x)μφ(d x)

and conversely, given a probability measure μ on any probability space Ω,

the above integral of the C-valued random variables on Ω is a positive linear

functional on the space. This kind of duality between functions (or random

variables) and the functionals on them (or measures or states) will be exploited

considerably in the sequel.

Given a probability space(Ω,�, μ

), or the linear space X of R-valued ran-

dom variables on it, let us now consider a map T : X → X with the properties

that

• Tf ∈ X+ (the cone of positive functions in X) if f ∈ X+, and

• T (1) = 1 (Markov map) or T (1) ≤ 1 (semi-Markov map).

For further mathematical analysis we shall have to put some topology on X,

for example, the weak-∗ topology on L∞R(Ω,�, μ), the real L∞(Ω,�, μ) space,

induced by its pre-dualX∗ = real L1(Ω,�, μ) := L1R(Ω,�, μ), or in a simplified

model where Ω is discrete, enumerated by N, say, X = l∞R

and X∗ = l1R, the

Banach spaces of real bounded and real summable sequences respectively. This

is the set up where we shall do the analysis and we shall be concerned with a

family or a semigroup of Markov maps :

T (t) : X → X, ∀ t ≥ 0 such that

the following conditions hold.

6.1. Probability and Markov semigroups 117

1. {T (t)}t≥0 is a C0-semigroup on X with respect to the locally convex

weak-∗ topology on X induced by X∗, that is,

(a) T (0)f = f and T (t)T (s)f = T (t+ s)f, ∀ t, s ≥ 0 and f ∈ X,

(b) 〈g, T (t)f〉 = ∫g(ω)(T (t)f)(ω)μ(dω) → ∫

g(ω)f(ω)μ(dω) as t → 0+

for all f ∈ X, and g ∈ X∗.

(c) However, note that weak-∗ continuity of the semigroup {T (t)}t≥0

on X is equivalent to the weak continuity of its pre-dual semigroup

T∗(t) on X∗. This, by Corollary 1.5.6 implies the strong continuity

of T∗(t) in X∗, making it a C0-semigroup (verify (Exercise 6.1.1)) in

the sense of Definition 2.2.5 and we shall treat it as such.

2. T (t)f ∈ X+ whenever f ∈ X+ ∀ t ≥ 0.

3. T (t)I = I ∀ t ≥ 0.

Such a semigroup is called a Markov semigroup. At this point it is instruc-

tive to point out that the convolution semigroups on probability measures, as

defined in Definition 2.5.6, give rise to a class of Markov semigroups as has been

proven in the Theorem 2.5.7. In view of condition (1c) above, it is convenient

analytically to deal with the pre-dual space ( for example L1 or l1 ) and the

pre-dual semigroup on it, rather than the semigroup on L∞ or l∞. Therefore

from this point onwards we shall set X = L1Ror l1

Rwith the C0 contraction

semigroup {T (t)}t≥0 on it and designate the dual semigroup as {T ∗(t)}t≥0 act-

ing on X∗ = L∞R or l∞

Rrespectively. In order to avoid confusion, the suffix R

shall henceforth be omitted while referring to these spaces. In the context of

the dual pair (l∞, l1) we collect a few elementary results, without proof, in the

next lemma.

Let X = l1 and X+ be the positive cone of l1. Then a linear map A : X →X is positive if A : D(A) ∩ X+ → X+ and we note that there is a normal

ordering of maps on l1 by this cone: A1 ≥ A2 if A1 − A2 is a positive map.

Note also that for any x, y ∈ X+, ‖x + y‖1 = ‖x‖1 + ‖y‖1. Then we have the

following.

Lemma 6.1.2. Let x, y, yj (j = 1, 2, . . .) belong to X+ and let A,B ∈ B(X) be

positive maps. Then the following are true.

(i) 0 ≤ A ≤ B implies that ‖A‖ ≤ ‖B‖;


(ii) 0 ≤ y1 ≤ y2 ≤ . . . and ‖yj‖1 ≤ M ∀ j implies that there exists a unique

y ∈ X+ such that ‖yj − y‖1 → 0 as j →∞;

(iii) 0 ≤ T1 ≤ T2 ≤ . . . with Tj ∈ B(X) such that ‖Tj‖ ≤ M ∀j implies

that there exists a unique positive T ∈ B(X) such that Tj converges to T

strongly.

The proof of the above lemma rests essentially on the monotone conver-

gence theorem for monotone sequences of reals, as well as for sequences in l1,

and is left as an exercise (Exercise 6.1.3).

6.2 Construction of Markov semigroups on a discrete state

space

To fix ideas, let us start with a finite state space Ω = {1, 2, . . . , N} and consider

a probability distribution given by a vector p = (p1, p2, . . . pN ) on it and a

stochastic (or Markov) matrix (tij)Ni,j=1 such that tij ≥ 0 and

∑Nj=1 tij = 1 for

each i.

(i) The probability vector p is in one-to-one correspondence with φp, the

positive linear functionals on the space of real-valued functions f on Ω

such that φp(f) =∑N

i=1 pif(i) and φp(1) = 1. This allows one to associate

a discrete evolution T given by T (φp)(χj) =∑

i pitij , where χj is the

indicator function of the singleton set {j}.(ii) There is a dual evolution T ∗ acting on the real-valued functions f on

Ω given by T ∗(f)(i) =∑

j tijf(j) and we observe that (T (φp))(χj) =

φp(T∗(χj)). It can also be noted that T ∗ maps positive functions to pos-

itive functions and maps the identity function to itself.

(iii) {T n}n∈Z+ and {T ∗n}n∈Z+ provide two discrete (dynamical) semigroups,

the second being dual to the first and clearly T ∗n satisfies property (ii)

for each n.

In probabilistic terminology, the element tij of the matrix representation of the

map T denotes the (Markov) transition probability from the initial state i to

the final state j, thereby explaining the requirements: tij ≥ 0 and∑N

j=1 tij = 1.

Now let N be the state space of a Markov transition over (continuous)

time t ∈ R+ with transition matrix P (t) ≡ (pij(t)), (i, j = 1, 2, . . .) such that

6.2. Construction of Markov semigroups on a discrete state space 119

it satisfies formally the ordinary differential equation (Chapman-Kolmogorov

equation):

d

dtpik(t) =

∞∑j=1

pij(t)ajk ord

dtP (t) = AP (t) (6.1)

where the matrix (pij(t)) acts on l1 as the right action (P (t)x)j =∑∞

i=1 xipij(t)

with initial conditions

limt→0+

P (t) ≡ P (0) = I, or equivalently limt→0+

pij(t) = δij . (6.2)

The map A appearing above is an N× N matrix (ajk) with the properties

ajk ≥ 0 for j �= k and∑∞

k=1 ajk = 0 for all j.

We also set aj ≡ −ajj so that aj =∑

k =j ajk ≥ 0.

}(6.3)

Intuitively, the condition (6.3), using (6.1), implies that

d

dt

∞∑k=1

pik(t) = 0,

that is,

d

dt‖P (t)x‖1 =

d

dt

∞∑j=1

∞∑i=1

xipij(t) = 0 for all x ∈ X+

or

‖P (t)x‖1 = ‖P (0)x‖1 = ‖x‖1 ∀x ∈ X,

where ‖x‖1 =∑∞

j=1 |xj |, the l1-norm. This means that the total probability

that the process, starting from the state i will transit to some state k ∈ N in

time t, is independent of t. Equivalently, we need to construct a C0-semigroup

{P (t)}t≥0 acting on l1 with generator A, which satisfies (6.3).

We remark here that Feller [9] first proved the existence of a unique mini-

mal semigroup associated with the equations (6.3) and (6.2) for infinite discrete

state space, and Kato [13] constructed the same in the framework of the the-

ory of semigroups, exploiting the special nature of l1. Here we follow Kato’s

construction.

As mentioned earlier, before the Lemma 6.1.2, we shall be dealing with

the Markov semigroup {P (t)}t≥0 acting on l1 and its dual {P ∗(t)}t≥0 on l∞.

Also note that the semigroup {P (t)}t≥0 or the operator A acts on the elements


of l1 by right action while its dual acts on the elements of l∞ by left action,

that is,

(Ax)j =

∞∑i=1

xiaij for x ∈ X and

(A∗y)j =∞∑k=1

ajkyk for y ∈ X∗.

Lemma 6.2.1. Let A be such that (6.3) holds. Then

(i) A is bounded if and only if supj aj <∞.

(ii) The natural domain of A is the maximal domain

D(A) := {x ∈ l1 :

∑k

∣∣∑j

xjajk∣∣ <∞}

.

Then A is a densely defined closed linear operator in X = l1.

(iii) Let D0 be the subspace ofX consisting of all sequences with finite support.

Then A0 ≡ A|D0 is called the minimal operator associated with A and is

also a closable densely defined operator. (Note that A0 need not be equal

to A.)

Proof. Let supj aj ≡ a <∞. Then, since

(Ax)k =∑j

xjajk = −xkak +∑j =k

xjajk,

we have that

‖Ax‖1 =∑k

|(Ax)k| ≤ a∑k

|xk|+∑k

∑j =k

|xj |ajk

= a‖x‖1 +∑j

|xj |∑k =j

ajk ≤ 2a‖x‖1,

showing that ‖A‖ ≤ 2a. Conversely, let supj aj = ∞. Then there exists a

subsequence {jl}, such that ajl ↑ ∞. Choose x(k)j = δjk so that ‖x(k)‖1 = 1 ∀k

and

‖Ax(k)‖1 =∑r

|∑j

x(k)j ajr| =

∑r

|akr| = |akk|+∑r =k

akr ≥ ak.

Therefore, ‖Ax(kl)‖1 ≥ akl↑ ∞ proving that A is not bounded. This proves

(i).


Since D0 ⊂ D(A) and D0 is dense in X, both A and A0 are densely defined

linear operators on X. Suppose {x(n)} ⊂ D(A) → x and Ax(n) → y ∈ X. This

means that for each j, r ∈ N,

|x(n)j − xj | ≤

∑l

|x(n)l − xl| → 0 and

∣∣∣∑j

x(n)j ajr − yr

∣∣∣ ≤∑k

|∑j

x(n)j ajk − yk| = ‖Ax(n) − y‖1 → 0

uniformly with respect to j and r respectively, as n→∞. Therefore,∣∣∣∑j

x(n)j ajk −

∑j

xjajk

∣∣∣ ≤ supj

∣∣x(n)j − xj

∣∣∑j

|ajk| ≤ 2ak supj

∣∣x(n)j − xj

∣∣→ 0

as n → ∞. Hence, yk =∑

j xjajk and by hypothesis y = (yk) ∈ X. Thus,

x ∈ D(A) and y = Ax. The closability of A0 may be proven similarly. �

Remark 6.2.2. If condition (i) of Lemma 6.2.1 above holds, then P ∗(t) = etA∗

and

d

dtP ∗(t)(I) = P ∗(t)A∗(I) = 0, that is,

∑ d

dtpik(t) =

∑j

pij(t)∑k

ajk = 0,

where I = (1, 1, 1, . . .), implying that P ∗(t)(I) = I, ∀ t ≥ 0, that is, P ∗ is a

Markov semigroup.

In view of the above lemma, two questions arise naturally :

Q1. What are the densely defined closed operators A in X such that A0 ⊂ A ⊂A and such that A is the generator of a positive contractionC0-semigroup?

Q2. Of all these semigroups (as possibly given as answer to Q1), which ones

are Markov; that is, for which ones does the following equality hold?

‖P (t)x‖1 = ‖x‖1 ∀x ∈ X.

Remark 6.2.3. (i) The property of conservativity: ‖P (t)x‖1 = ‖x‖1 for all

x ∈ X is equivalent to ‖P (t)x‖1 = ‖x‖1 for all x ∈ X+, as well as to

P ∗(t)(I) = I for all t ≥ 0, where P ∗ is the adjoint semigroup acting on

l∞ = X∗ and I = (1, 1, 1 . . .). The first equivalence follows from the facts

that P is a positive map and the property of the l1-norm, and the second

from the duality between l1 and l∞, that is, 〈P ∗(t)y, x〉 = 〈y, P (t)x〉 forall x ∈ X and y ∈ X∗.


(ii) In particular, if we choose x = ej , the canonical basis (the sequence

(ej)k = δjk) in X = l1, then (P (t)ej)k =∑

plk(t)(ej)l = pjk(t) > 0,

and therefore, the requirement of conservativity, ‖P (t)ej‖1 = ‖ej‖1 = 1

implies that∑

k pjk(t) = 1 ∀j, while the contractivity of P implies that∑k pjk(t) ≤ 1 ∀j.

So the problem at hand is to answer Q1 and Q2 given earlier, that is, to

construct a contraction semigroup P with generator A in X = l1, given the

conditions (6.3), and also to determine the necessary and sufficient conditions

for the constructed semigroups P to be conservative or Markov.

Let ajk and aj ≡ −ajj be as in (6.3) and define a positive (possibly

unbounded) operator H in X by

D(H) = {x ∈ X :∑

j |ajxj | <∞} and (6.4)

(Hx)j = ajxj ∀ x ∈ D(H). (6.5)

Then clearly, D0 � D(H). We set D+ = D(H) ∩ X+ and (D0)+ = D0 ∩ X+

leading to the next lemma.

Lemma 6.2.4. (i) H defined above, is a positive, closed operator in X with

D0 a core.

(ii) For all λ > 0, (λ+H)−1 is a positive, bounded operator in X with

‖(λ+H)−1‖ ≤ λ−1.

(iii) If we define a linear operator K in X with domain D(H) by setting

(Kx)k =∑j =k

xjajk,

then K is a positive operator, bounded relative to H, that is, ‖Kx‖ ≤‖Hx‖ for all x ∈ D(H) (see Definition 4.1.2), and ‖Kx‖ = ‖Hx‖ ∀ x ∈D+.

(iv) Ax = −Hx+Kx for any x ∈ D(H).

The proof is omitted since it is straightforward, except to note that the

contraction semigroup e−tH leaves D0 invariant and that fact, by Theorem

A.1.9, implies that D0 is a core for H. The details are left for the reader to

verify (Exercise 6.2.5).


Remark 6.2.6. (i) D(A) is in general strictly bigger than D(H) and

D(H) = D(−H +K), that is, A ⊃ −H +K ⊃ A0 = A|D0 .

(ii) Since, in general, the bound of K relative to H is not less than 1, none

of the standard results in the theory of perturbation of contraction semi-

groups (Theorems 4.1.3 and 4.1.5, and Corollary 4.1.4) are applicable here,

though the unperturbed semigroup T (t) = exp−tH is a positive contrac-

tion semigroup. The strategy, (following Feller and Kato) is to capitalise

on the positivity property rather than the small relative bound property.

For this, we first set Gr := −H + rK for 0 ≤ r < 1 and collect some of

the relevant results in the next lemma.

Lemma 6.2.7. (i) B(λ) := K(λ + H)−1, with K and H as in Lemma 6.2.4,

is a positive contraction in X for each λ > 0, satisfying 0 ≤ B(μ) ≤ B(λ)

for 0 < λ < μ.

(ii) For each 0 ≤ r < 1, Gr is the generator of a contraction C0-semigroup Pr

and its resolvent Rr(λ) := (λ−Gr)−1 is a positive bounded operator with

‖Rr(λ)‖ ≤ λ−1 for all λ > 0.

Proof. (i) That B(λ) is contractive follows from Lemma 6.2.4,(iii), and that

it is positive follows from the positivity of K and of (λ +H)−1, and the

fact that composition of two positive operators is positive. Further, by the

resolvent equation, for 0 < λ < μ, B(λ)−B(μ) = (μ− λ)B(λ)(μ+H)−1

is positive.

(ii) It is easy to see that

Rr(λ) = (λ+H − rK)−1 = (λ+H)−1(1 − rB(λ))−1

=

∞∑n=0

rn(λ+H)−1B(λ)n

with the Neumann series converging in operator norm since ‖B(λ)‖ ≤ 1.

This also proves that Rr(λ) is a positive map in X. For f ∈ D+, and

λ > 0,

‖(λ−Gr)f‖ = ‖(λ+H)f − rKf‖ ≥ ‖(λ+H)f‖ − r‖Kf‖= ‖λf‖+ (‖Hf‖ − r‖Kf‖) ≥ λ‖f‖

since ‖Kf‖ ≤ ‖Hf‖.


Furthermore, since Rr(λ) = (λ−Gr)−1 is a positive bounded map,

(λ−Gr)D+ = X+.

Thus for g ∈ X+,

‖Rr(λ)g‖ = ‖f‖ where (λ −Gr)f = g for some f ∈ D+.

Hence ‖Rr(λ)g‖ = ‖f‖ ≤ λ−1‖g‖. For any g ∈ X, g = g+ − g− and

‖g‖ = ‖g+‖+ ‖g−‖ with g± ∈ X+, leading to

Rr(λ)g = Rr(λ)g+ −Rr(λ)g−

and

‖Rr(λ)g‖ = ‖Rr(λ)g+‖+ ‖Rr(λ)g−‖ ≤ λ−1(‖g+‖+ ‖g−‖) = λ−1‖g‖.

While Theorem 4.2.1 shows that Gr is a closed operator for 0 ≤ r < 1, an

application of the Hille-Yosida Theorem 2.3.3 (i) proves that the operator

Gr = −H + rK is the generator of a contraction C0-semigroup on X.

�

If we denote by {Pr(t)}t≥0 the contraction semigroup generated by Gr for

0 ≤ r < 1, then we have the following main result.

Theorem 6.2.8. For 0 ≤ r < 1, the contraction semigroup {Pr(t)}t≥0 satisfies

the following.

1. Pr(t) is positive for every t ≥ 0.

2. {Pr(t)} converges strongly to a positive contraction C0-semigroup {P (t)}as r increases to 1 and the strong convergence is uniform for t in any

compact subset of [0,∞).

Proof. Fix λ > 0. Then the map (0, 1) � r �→ Rr(λ) is positive and increasing,

since for r > s,

Rr(λ)− Rs(λ) = (λ+H)−1∞∑n=0

(rn − sn)B(λ)n

is positive. Now, by Remark 2.3.4

Pr(t) = s− limn→∞

[ntRr

(nt

)]n


which implies that each Pr(t) is positive and also that for r > s,

Pr(t)− Ps(t)

= s− limn→∞

[(nt

)n n−1∑j=0

Rr

(nt

)n−1−j[Rr

(nt

)−Rs

(nt

)]Rs

(nt

)j]is positive. Thus, for a fixed t ≥ 0, the family of contractions [0, 1) � r �→ Pr(t)

is monotonically increasing and therefore, by Lemma 6.1.2, Pr(t) converges

strongly, to say P (t), as r ↑ 1. This implies that {P (t)}t≥0 is a contractive

positive family and Pr(t) ≤ P (t) for all t ≥ 0 and 0 ≤ r < 1. That {P (t)}t≥0

is a semigroup follows from strong convergence and the identity:

Pr(t+ s) = Pr(t)Pr(s) for all r ∈ [0, 1) and t, s ≥ 0.

Similarly we conclude that P (0)f = limr Pr(0)f = f for all f.

Finally we show that the strong convergence of Pr(t) to P (t) as r ↑ 1 is

uniform for t in a bounded interval I and as a consequence of this, also prove

the C0 property. If this is not true, then there exist η > 0 and sequences {rn}and {tn} such that tn → t0 ∈ I and rn ↑ 1, with the property that

‖P (tn)f − Prn(tn)f‖ ≥ η > 0

for some f ∈ X+. In such a case, since both Prn(tn) and [P (tn)− Prn(tn)] are

positive maps, one has that

‖P (tn)f‖ = ‖Prn(tn)f‖+ ‖[P (tn)− Prn(tn)

]f‖ ≥ ‖Prn(tn)f‖+ η

which yields that for m < n (with rm < rn),

‖Prm(tn)f‖ ≤ ‖Prn(tn)f‖ ≤ ‖P (tn)f‖ − η.

Now, letting n→∞, so that rn ↑ 1, and tn → t0, we have that

‖Prm(t0)f‖ ≤ ‖P (t0)f‖ − η which implies ‖[P (t0) − Prm(t0)]f‖ > η and this

contradicts the earlier proven fact that Prm(t0) converges strongly to P (t0) as

m→∞. This implies that for f ∈ X,

limt↓0

P (t)f = limt↓0

limr↑1

Pr(t)f = limr↑1

limt↓0

Pr(t)f = f = P (0)f,

showing that {P (t)}t≥0 is a C0 semigroup.

�


The next theorem studies the properties of the limit semigroup P in detail;

in particular it describes in the precise sense in which P is the minimal of all

semigroups constructed out of the formal sum −H +K.

Theorem 6.2.9. Let G be the generator of the semigroup P constructed in the

previous theorem and let R(λ) = (λ − G)−1 for λ > 0 be its resolvent. Then

the following hold.

(i) ‖R(λ)‖ ≤ λ−1, 0 ≤ Rr(λ) ≤ R(λ) for 0 ≤ r < 1 and Rr(λ) converges

strongly to R(λ) as r ↑ 1, for each λ > 0.

(ii) Set R(n)(λ) = (λ + H)−1∑n

j=0 B(λ)j for λ > 0. Then R(n)(λ) ≤ R(λ)

and R(n)(λ) converges strongly to R(λ) as n→∞.

(iii) G is a closed extension of −H+K on D(H) and hence also of A0 = A|D0 .

Furthermore, G is a restriction of A, that is, A ⊃ G ⊃ −H +K ⊃ A0.

(iv) (minimality) If there exists a positive C0-semigroup {P ′(t)}t≥0 with gen-

erator G′ which extends A0, then P ′(t) ≥ P (t) for all t ≥ 0.

Proof. (i) We have seen that R(λ) =

∫ ∞

0

e−λtP (t) dt (λ > 0), by Theorem

2.2.7(c) and similarly,

Rr(λ) =

∫ ∞

0

e−λtPr(t) dt (λ > 0).

It is immediately clear that the positivity of the semigroups implies the

same property for the resolvents and the norm bounds follow from Theo-

rem 2.2.7(f). Note that for f ∈ X,

R(λ)f −Rr(λ)f =

∫ ∞

0

e−λt(P (t)f − Pr(t)f) dt

and therefore

‖R(λ)f −Rr(λ)f‖ ≤∫ α

0

e−λt‖P (t)f − Pr(t)f‖ dt+ 2‖f‖∫ ∞

α

e−λt dt.

Given an ε > 0, choose α > 0 such that 2‖f‖∫ ∞

α

e−λt dt <ε

2and for

that α, choose δ > 0, with 1− δ < r < 1, so that

‖P (t)f − Pr(t)f‖ < ε

2∀t ∈ [0, α],

by using the result on uniform convergence in Theorem 6.2.8. This will

imply that for such a choice of r, ‖R(λ)f −Rr(λ)f‖ < ε, completing the

proof.


(ii) Set

R(n)r (λ) = (λ+H)−1

n∑m=0

rmB(λ)m.

It is clear from the definition and part (i) above that

(1) 0 ≤ R(n)r (λ) ≤ Rr(λ) ≤ R(λ) for all n ∈ N,

(2) 0 ≤ R(m)r (λ) ≤ R(n)

r (λ) for m ≤ n and that

(3) R(n)r (λ) converges strongly to

R(n)(λ) = (λ +H)−1n∑

j=0

B(λ)j ,

a positive bounded operator as r ↑ 1.These imply that 0 ≤ R(m)(λ) ≤ R(n)(λ) ≤ R(λ) for λ > 0 and for

n ≥ m. Thus, by Lemma 6.1.2(iii), R(n)(λ) increases strongly to a bounded

operator, say R(λ) and R(λ) ≤ R(λ). On the other hand, R(n)r (λ) ≤

R(n)(λ) ≤ R(λ) and taking strong limit in the left hand side of this

inequality, first as n → ∞, and then as r ↑ 1, we get that R(λ) ≤ R(λ),

leading to the required result, which means that

R(λ) = (λ+H)−1∞∑n=0

B(λ)n,

the infinite sum converging strongly.

(iii) Note that

R(n)(λ) = (λ +H)−1 + (λ+H)−1( n−1∑

j=0

B(λ)j)B(λ)

= (λ +H)−1 +R(n−1)(λ)B(λ)

and therefore, for f ∈ D(H) = D(K), we have that

R(n)(λ)(λ +H)f = f +R(n−1)(λ)Kf.

Now letting n→∞ we get that

R(λ)(λ +H)f = f +R(λ)Kf, or equivalently,

R(λ)(λ +H −K)f = f for all f ∈ D(H).


This implies that f ∈ D(G), that is, D(H) ⊂ D(G), and for such f, (λ −G)f = (λ+H−K)f, so that G = −H+K on D(H) or G ⊃ −H+K. Thus

G is a closed extension of −H +K. Since RanR(n)(λ) ⊂ D(H) = D(K),

we get that for f ∈ X+,

(λ+H)−1KR(n)(λ)f =

n+1∑j=1

(λ+H)−1B(λ)jf = R(n+1)(λ)f−(λ+H)−1f.

(6.6)

The right hand side in (6.6) converges to R(λ)f − (λ+H)−1f as n→∞.

Since l1-convergence implies pointwise convergence (choosing a subse-

quence if necessary), this means that the left hand side vector in (6.6),

represented by the sequence

(λ+ ak)−1

∑j =k

ajky(n)j , converges to (λ + ak)

−1∑j =k

ajkyj ,

where

y(n)j =

(R(n)(λ)f

)j, yj =

(R(λ)f

)jand ajk > 0 (j �= k).

This observation and (6.6) implies that

(λ+ ak)−1

∑j =k

ajkyj = yk − (λ+ ak)−1fk

and thus

(Ky)k = (λ+ ak)yk − fk =((λ+H)y

)k− fk or

fk = ((λ+H −K)y)k = λyk −∞∑j=1

yjajk.

For any f ∈ X, we write f = f+ − f− and conclude similarly that

(f+)k = ((λ+H −K)y+)k and (f−)k = ((λ +H −K)y−)k,

where y± = R(λ)f±. This gives the result that RanR(λ) ⊂ D(A) and

(λ+H −K)R(λ)f = f for all f ∈ X. Consequently,

A ⊃ G ⊃ −H +K ⊃ A0.


(iv) Let G′ be the generator of another C0-semigroup P ′ extending −H +K.

Then for x ∈ D(H), there exists a sequence {xn} ⊂ D0 such that xn → x

and Hxn → Hx since D0 is a core for H by Lemma 6.2.4(i), which implies

that

‖K(x− xn)‖ ≤ ‖H(x− xn)‖ → 0 as n→∞.

This leads to the conclusion that

A0xn = (−H +K)xn → (−H +K)x as n→∞.

This fact along with the assumption of G′ being a closed extension of A0

implies that

D(G′) ⊃ D(H) and G′x = (−H +K)x for all x ∈ D(H),

that is, G′ ⊃ −H +K. If the C0-semigroup P ′ is of type β, then

{λ ∈ R : λ > β} ⊂ ρ(G′)

and we set R′(λ) = (λ −G′)−1. Then we have that

R′(λ)−Rr(λ) = R′(λ)(G′ −Gr)Rr(λ) = (1− r)R′(λ)KRr(λ),

which is positive since R′(λ), the resolvent of a positive semigroup P ′, is

also positive for λ > λ0 := max(0, β) and since 0 ≤ r < 1. This implies

that, Rr(λ) ≤ R′(λ), leading to R(λ) ≤ R′(λ), for each λ > λ0. Fix t > 0,

and note that for all large enough n ∈ N, n/t > λ0. Therefore,

n

t

(nt−G′

)−1

≥ n

t

(nt−G

)−1,

which implies that[nt

(nt−G′

)−1]n−[nt

(nt−G

)−1]n=(nt

)n n−1∑j=0

(nt−G′

)−(n−j−1)[(nt−G′

)−1

−(nt−G

)−1](nt−G

)−j

(6.7)

is positive. Taking strong limit as n → ∞ on the left hand side of (6.7),

as in the proof of Theorem 6.2.8, we get that

P ′(t) ≥ P (t) ∀t > 0.

Note that for t = 0, P ′(0) = I = P (0).

�


The contractive C0-semigroup {P (t)}t≥0 constructed in Theorem 6.2.8 is

thus naturally called the minimal semigroup.

As explained earlier, unlike in the case of a finite state space, or equiv-

alently, unlike when the underlying space of analysis is finite-dimensional, in

the more general case, the condition (6.2) that∑∞

k=1 ajk = 0 for all j, does

not necessarily imply the conservation of probability, that is, ‖P (t)f‖1 may

not be equal to ‖f‖1 for all f ∈ X, and t > 0, or for the dual semigroup on

l∞, P ∗(t)I �= I for all t > 0, where I is the sequence (1, 1, 1, . . .) ∈ l∞. Next,

the necessary and sufficient conditions for the conservativity or the Markov

property of the minimal semigroup P are studied.

Lemma 6.2.10. (i) For all x ∈ X+ and λ > 0, b(λ;x) ≡ limn→∞ ‖B(λ)nx‖exists, and defines the map b(λ, ·) as a map from the positive cone X+ into

the cone R+, such that b(λ;αx+ y) = αb(λ;x) + b(λ; y) for all x, y ∈ X+

and α > 0.

(ii) For any fixed x ∈ X+, either b(λ;x) = 0 for all λ > 0 or b(λ;x) > 0 for

all λ > 0. Also, b(λ;x) is a non-increasing function of λ and limλ↓0 b(λ;x)

exists.

(iii) For any x ∈ X+, the function [0,∞) � t �→ ‖P (t)x‖ is a non-increasing

function and limt→∞ ‖P (t)x‖ = ‖x‖ − limλ↓0 b(λ;x).

(iv) Furthermore, for any λ > 0, and x ∈ X+,

(0,∞) � λ �→ λ−1[‖x‖− b(λ;x)] ∈ (0,∞] is also a non-increasing function

and ∫ ∞

0

‖P (t)x‖ dt = limλ↓0

λ−1[‖x‖ − b(λ;x)],

which may be +∞.

Proof. (i) Fix λ > 0, and note that

I +KR(n)(λ) =n+1∑j=0

B(λ)j = (λ+H)R(n)(λ) +B(λ)n+1, (6.8)

and thus for x ∈ X+,

‖x‖+ ‖KR(n)(λ)x‖ = λ‖R(n)(λ)x‖ + ‖HR(n)(λ)x‖ + ‖B(λ)n+1x‖,

where we have used the fact that for x, y ∈ X+, ‖x+ y‖ = ‖x‖+ ‖y‖.


Since R(n)(λ)x ∈ D+, we have by Lemma 6.2.4 (iii) that

‖KR(n)(λ)x‖ = ‖HR(n)(λ)x‖

and it follows that

‖B(λ)n+1x‖ = ‖x‖ − λ‖R(n)(λ)x‖.

Therefore, by Theorem 6.2.9(ii), b(λ;x) exists for every x ∈ X+ and

b(λ;x) = ‖x‖ − λ‖R(λ)x‖ = λ

∫ ∞

0

e−λt(‖x‖ − ‖P (t)x‖) dt, (6.9)

by Theorem 2.2.7(e). The final claim follows immediately from (6.9).

(ii) If b(λ;x) = 0 for some λ > 0, then it follows from (6.9) that for almost

all t, ‖x‖ − ‖P (t)x‖ = 0, and therefore, b(λ;x) = 0 for all λ > 0. That

b(λ;x) if not identically zero, is non-increasing in λ follows from the same

property of B(λ) (see Lemma 6.2.7(i)) and their positivity. The existence

of limλ↓0 b(λ;x) follows from the fact that it is a non-increasing function

of λ.

(iii) That ‖P (t)x‖ is a non-increasing function of t follows from the contraction

semigroup property of P ; this and (6.9) leads to

‖x‖ − b(λ;x) = λ

∫ ∞

0

e−λt‖P (t)x‖ dt.

Letting λ ↓ 0, in above gives that limλ→0

b(λ;x) = ‖x‖ − limt→∞

‖P (t)x‖,(the reader may verify this equality (Exercise 6.2.11) ) showing that

limt→∞

‖P (t)x‖ exists for x ∈ X+.

(iv) This follows in a manner similar to above and the details are left for the

reader to work out (Exercise 6.2.12).

�

Remark 6.2.13. (i) For x ∈ X+, ‖P (t)x‖ = ‖x‖ if and only if b(λ;x) = 0 for

some, and hence for all λ > 0, or equivalently if and only if B(λ)nx→ 0 as

n→∞. This follows clearly from the proof of Lemma 6.2.10, in particular

from (6.9).

(ii) In the theory of perturbation of semigroups or of the generators of semi-

groups, it is the smallness of the perturbation relative to the unperturbed


generator in some sense, that is used. Here, the positive operator H is the

generator of the unperturbed semigroup and K is the perturbation. But

as noted in Lemma 6.2.4(iii), K is not small relative to H and hence the

methods of Chapter 4 are not available. However, the fact that both H

and K are positive maps comes to our rescue. Nevertheless, a price has

to be paid in the sense that the uniqueness of the perturbed semigroups

is lost and one needs to concentrate on the minimal one only.

(iii) If we choose, in particular, x = ej , the j-th member of the canonical basis

ofX, then x ∈ X+ and b(λ, ej) = 0 if and only if ‖P (t)ej‖ =∑

k pjk(t) = 1

for all t and each j. This follows from (6.9).

The next theorem collects all these results relevant to the conservativity

of the minimal semigroup.

Theorem 6.2.14. The minimal semigroup P is conservative (or Markov) that

is, ‖P (t)x‖ = ‖x‖ ∀x ∈ X+, if and only if any one of the following equivalent

conditions is satisfied.

(i) For some λ > 0, and hence for all λ > 0 and all x ∈ X+,

b(λ;x) ≡ limn→∞ ‖B(λ)nx‖ = 0.

(ii) For some λ > 0, the dual eigenvalue equation A∗0x

∗ = λx∗ has no non-zero

solution in X∗ = l∞.

(iii) For some λ > 0, Ran(λ−A0) is dense in X where A0 = A|D0 .

Furthermore, when any of the above conditions (and hence all) is satisfied, then

the minimal semigroup P is the only semigroup, the generator of which is an

extension of A0.

Proof. In view of Remark 6.2.13(i), it is sufficient to prove the equivalence of

the statements (i), (ii) and (iii). That (ii) is equivalent to (iii) follows by using

the equality

N ((A0 − λ)∗) = Ran(A0 − λ)⊥,

where N (T ) is the null space of the operator T and Ran(T )⊥ is the annihilator

of the range of T, together with an application of the Hahn-Banach theorem.

(Exercise 6.2.15)

(i) ⇔ (iii). From (6.8) one has that

(λ +H −K)R(n)(λ)x = x−B(λ)n+1x,


and therefore (i) implies that (λ+H −K)R(n)(λ)x converges strongly to x as

n→∞ for every x ∈ X. Since D(H) ⊃ R(n)(λ)x converges to R(λ)x as n→∞and since RanR(λ) = D(H), this means that

Ran(λ+H −K) is dense in X.

Since by Lemma 6.2.4, D0 is a core for H , and since for x ∈ D0 ⊂ D(H),

‖Kx‖ ≤ ‖Hx‖, it follows that

(λ+H −K)D(H) = (λ +H −K)D0.

Thus we have that (λ −A0)D0 = X .

Conversely, let (λ−A0)D0 = X . Then, since (λ+H)D(H) = X , one has that

(I −B(λ))X = [I −B(λ)](λ +H)D(H) = (λ+H −K)D(H) ⊇ (λ−A0)D0,

which is assumed to be dense and thus Ran[I − B(λ)] is dense in X. Set

Bn(λ) =1

(n+ 1)

n∑j=0

B(λ)j

so that for every x ∈ X,

Bn(λ)[I −B(λ)]x = (n+ 1)−1[x−B(λ)n+1x],

which converges to 0. Since Ran[I − B(λ)] is dense and ‖Bn(λ)‖ ≤ 1 ∀n,it then follows that Bn(λ)x → 0 for as n −→ ∞ for any x ∈ X. Using the

contractivity of B(λ), one notes that

‖B(λ)nx‖ ≤ ‖B(λ)jx‖ if 0 ≤ j ≤ n.

Hence for x ∈ X+,

‖Bn(λ)x‖ = 1

n+ 1

n∑j=0

‖B(λ)jx‖ ≥ ‖B(λ)nx‖

and one concludes that

b(λ;x) = limn−→∞ ‖B(λ)nx‖ = 0 ∀x ∈ X+.

This completes the proof. �


To end the chapter, we discuss a few concrete examples with infinite state

space.

Example 6.2.16. (The bounded case, i.e., sup |aj | ≡ a <∞)

We have already seen that since this implies that H and K are both bounded

positive operators, there is only one (uniformly continuous) semigroup P and

that is clearly Markov. However, we discuss this now with Theorem 6.2.14 as

the backdrop. Note that∥∥B(λ)x∥∥ =

∥∥K(λ+H)−1x∥∥ ≤ ∥∥H(λ+H)−1x

∥∥=∑j

∣∣( aj(λ+ aj)

)xj

∣∣ ≤ ( a

(λ+ a)

)‖x‖implying that

∥∥B(λ)∥∥ ≤ a(λ + a)−1 and therefore

∥∥B(λ)n∥∥ −→ 0 as n −→ ∞

for every λ > 0.

Example 6.2.17. (The death process) Here assume that the infinite matrix

A is lower triangular, that is, ajk = 0 ∀ k > j, and therefore(B(λ)x

)k

=∑j =k

xjbjk(λ) with the matrix element bjk(λ) = 0 if k > j and = ajk/λ+ aj if

k < j. Then(B(λ)nej

)k=∑( ajnjn−1

λ+ ajn

)( ajn−1jn−2

λ+ ajn−1

)· · · ·

( aj2j1λ+ aj2

)( aj1kλ+ aj1

),

where the summation above is taken over the n indices j1, j2, . . . , jn, subject

to the conditions j = jn �= jn−1, jn−1 �= jn−2, . . . j2 �= j1, j1 �= k. By virtue of

the assumption on the matrix elements ajk, the above sum is non-zero only if

j = jn ≥ jn−1 + 1 ≥ jn−2 + 2 ≥ ... ≥ j1 + n − 1 ≥ k + n. This implies that(B(λ)nej

)k= 0 if j < k + n and for each fixed j,

∥∥B(λ)nej∥∥ =

∞∑k=1

(B(λ)nej

)k= 0

if we choose n large enough. Therefore, for x =∞∑j=1

αjej, since B(λ) is a con-

traction,

∥∥B(λ)nx∥∥ ≤ ∥∥B(λ)n

N∑j=1

αjej∥∥+

∥∥x− N∑j=1

αjej∥∥,

and we have that B(λ)n converges strongly to 0. Equivalently, by Theorem

6.2.14, the associated minimal semigroup is Markov.


Example 6.2.18. (The birth process) In this example, the matrix ajk has the

property that aj,j+1 = −ajj = aj ≥ 0 and all other ajk = 0. This means that

for the matrix(bjk(λ) ≡

(B(λ)ej

)k

)all bjk(λ) = 0 except for bj,j+1(λ) which

is equal toaj

λ+aj. Therefore,

(B(λ)nej

)k= 0 unless k = j + n and

∥∥B(λ)nej∥∥ =

(B(λ)nej

)j+n

=

n−1∏m=0

( aj+m

λ+ aj+m

)

=

n−1∏m=0

(1 + λa−1

j+m

)−1.

For fixed j, the infinite product above converges to 0 as n → ∞ if and only

if∞∑

m=0

log(1 + λa−1

j+m

)= ∞, which happens if and only if either aj = 0 for

infinitely many j or∞∑

j=N

aj−1 =∞ for some N such that aj > 0 for all j ≥ N .

On the other hand if∞∑

m=0a−1j+m <∞ for some j, then

bj(λ) ≡ limn−→∞

∥∥B(λ)nej∥∥ =

∞∏m=0

(1 + λa−1

j+m

)−1

= 1− λ

∞∑m=0

a−1j+m + o(λ2) for small λ > 0.

In such a case, by Lemma 6.2.10(iii),

limt−→∞

∥∥P (t)ej∥∥ =

∥∥ej∥∥− limλ↓0

bj(λ) = 0, that is, limt−→∞

∑k

pjk(t) = 0.

It also follows from Lemma 6.2.10(iv), that∑k

∫ ∞

0

pjk(t)dt =

∫ ∞

0

∥∥P (t)ej∥∥dt = lim

λ↓0{λ−1

(∥∥ej∥∥− bj(λ))}

=

∞∑m=0

a−1j+m <∞.

Remark 6.2.19. Many results, similar to the ones derived in this chapter, can

be obtained also in the non-commutative context. For example, one can study

the Markov or semi-Markov semigroups on B(H), or its pre-dual semigroup

on B1(H) [6]. A more general extension of this theory to an arbitrary von

Neumann algebra or its pre-dual Banach space, respectively, can be found in

[25, Section 3.2].

Chapter 7

Applications to partial differential

equations

While semigroups of operators are interesting objects of study by themselves,

one of the main reasons why they are studied so extensively is due to the

important role they play in the study of partial differential equations.

Consider the well-known initial value problem:⎧⎨⎩

∂u

∂t(x, t) = Δu(x, t), x ∈ Rd, t ≥ 0

u(x, 0) = f(x), x ∈ Rd,(7.1)

where

Δu =

d∑i=1

∂2u

∂x2i

.

Finding solutions of the above using the method of separation of variables,

which is an effective method only in the presence of some symmetry in the sys-

tem, is part of every basic course on partial differential equations, often leaving

many fundamental issues like uniqueness, completeness and so on unanswered.

However, the theory of C0-semigroups can be employed to treat an equation

such as this in an elegant manner, yielding, in most cases pay-offs beyond just

existence and uniqueness of solutions. To begin with, given the linear partial

differential equation, one tries to write it as an abstract Cauchy problem on a

Banach space X , {u′(t) = Au(t), t ≥ 0

u(0) = f(7.2)


138 Applications to partial differential equations

where the space X is a suitably chosen concrete Banach space of functions in

which the initial value lies and A is the differential operator (from the given

partial differential equation) with an appropriate domain D(A) ⊂ X. The solu-

tion of the given differential equation is then a trajectory in X. Depending on

the nature and properties of the operator A, the trajectory may be described

by a C0-semigroup, which is a very convenient tool. Indeed, if the operator

A is the infinitesimal generator of a C0-semigroup on X, say {T (t)}t≥0, then

u(t) = T (t)f gives mild (f ∈ X) or classical (f ∈ D(A)) solutions of the Cauchyproblem and hence of the original problem.

We now discuss some simple differential operators and show that they

generate C0-semigroups on appropriate function spaces.

7.1 Parabolic equations

By a multi-index α we shall mean that α = (α1, α2, . . . , αd) where α1, . . . , αd ∈N ∪ {0}. We set |α| = ∑j=d

j=1 αj , and xα = xα11 xα2

2 · · ·xαd

d , for x ∈ Rd and

Dα = Dα1

1 · · · Dαd

d , where Dj = −i ∂∂ xj

. If P is the polynomial defined by

P (x) =∑

|α|≤m aαxα where aα ∈ R, then we set P (D) =

∑|α|≤m aαD

α. Note

that in contrast to the notation Dj and Dα introduced in Section A.2, here Dj

has a factor −i in front of∂

∂ xj. This is done to avoid the appearance of the

powers of i in the polynomial P in the Fourier variable (see Lemma A.2.3).

If P is a polynomial, then P (D) is a linear differential operator with con-

stant coefficients . The order of P (D) is the degree of the polynomial. The

polynomial P is called homogeneous if it is of the form P (x) =∑

|α|=m aαxα.

We shall call a homogeneous polynomial of degree 2m weakly elliptic if P (x) ≤ 0

and strictly elliptic if P (x) < 0 for all x ∈ Rd \ {0}. Note here that a non-zero

homogeneous polynomial satisfying P (x) ≤ 0, for all x ∈ Rd \{0} is necessarilyof even order. If P is strictly elliptic of order 2m, then P is non positive on the

unit sphere of Rd, which is compact. It follows therefore that there is a constant

θ > 0 such that P (x) ≤ −θ|x|2m for all x ∈ Rd, where |x|2 = x21 + · · ·+ x2

d.

7.1. Parabolic equations 139

From the discussions in Appendix A.2 it follows that P (D) has a natural

extension as an operator acting on the space L2(Rd). Precisely, we have that

D(P (D)) = {f ∈ L2(Rd) : P (·)Ff ∈ L2(Rd)},P (D) = F−1MPF ,

where MP is the multiplication operator (see Appendix A.1 and also Section

2.5) given by

(MP f)(x) = P (x)f(x), for all x ∈ Rd,

for all f ∈ D(MP ) ={f ∈ L2(Rd) :

∫Rd

|P (x)f(x)|2 dx <∞}.

Thus MP and P (D) are unitarily equivalent.

If Q(x, y) =∑

bα(x)yα is a polynomial in y ∈ Rd such that the coefficients

bα are functions of x ∈ Rd, then the operator Q(·, D) given by

(Q(·, D)f)x =∑

bα(x)Dαf(x)

is called a linear partial differential operator with variable coefficients. Note

here that Q(·, D) can be written as Q(·, D) =∑

MbαDα.

Theorem 7.1.1. If P is a weakly elliptic homogeneous polynomial of degree

2m (m = 1, 2, . . .) with real coefficients, then P (D) generates a holomorphic

semigroup on L2(Rd) on the sector Re z > 0 (that is, of angle π/2 ) and a

unitary group for Re z = 0. Furthermore, if P is strictly elliptic and Q(x, y) =∑|α|≤2m−1 bα(x)y

α, where bα ∈ L∞(Rd), for all α, then P (D) + Q(·, D) also

generates a holomorphic semigroup.

Proof. Since P is weakly elliptic, it is in particular real valued and therefore,

from Theorem A.1.13, it follows that MP and hence P (D) is selfadjoint. More-

over, P (D) ≤ 0, since P being weakly elliptic implies

〈MP f, f〉L2(Rn) =

∫Rn

P (x)|f(x)|2 dx ≤ 0 ∀ f ∈ D(MP ),

that is, MP ≤ 0. Therefore, by Example 3.4.2, it follows that P (D) generates

a holomorphic semigroup of angle π/2. By Stone’s Theorem, iP (D) generates

a unitary C0-group on R or equivalently, P (D) generates a unitary C0-group

for Re z = 0.


We shall prove that Q(·, D) =∑

MbαDα is relatively bounded with re-

spect to P (D), in fact that its P (D)-bound is less than 1. Let α be a multi-index

with |α| ≤ 2m− 1. Since the map Rd � y �→ yα is continuous where | · | is theEuclidean norm in Rd, and since

∣∣|yα|/|y|2m∣∣ = |y||α|−2m converges to 0 as

|y| → ∞, we conclude that for every ε > 0, we can find cε > 0 such that

|yα|2 ≤ c2ε + ε2|y|4m ∀ y ∈ Rd.

Since P is strictly elliptic, there exists θ > 0 such that P (y) ≤ −θ|y|2m for all

y ∈ Rd, so |P (y)|2 ≥ θ2|y|4m. Therefore, by the results in Appendix A.2, and

in view of the discussion immediately preceding Theorem 7.1.1, it follows, on

setting φα(y) = yα, that

‖Dαf‖2L2(Rd) = ‖F−1MφαFf‖2L2(Rd) =

∫Rd

|(MφαFf)(y)|2 dy

=

∫Rd

|yα|2|Ff(y)|2 dy

≤ c2ε‖Ff‖2 + ε2∫Rn

|y|4m|F(y)|2 dy

≤ c2ε‖f‖2 +ε2

θ2‖P (D)f‖2 (7.3)

≤ (cε‖f‖+ ε

θ‖P (D)f‖)2. (7.4)

Since bα ∈ L∞(Rd),Mbα is a bounded operator on L2(Rd), and ‖Mbα‖ ≤‖bα‖∞. Therefore, using (7.4), we get, for f ∈ D(P (D)),

‖Mbα(Dαf)‖ ≤ ‖bα‖∞‖Dαf‖ ≤ ε‖bα‖∞θ−1‖P (D)f‖+ ‖bα‖∞cε‖f‖.

This holds for every α with |α| ≤ 2m− 1. Therefore,

‖Q(·, D)f‖ =∥∥∥ ∑

|α|≤2m−1

MbαDαf∥∥∥ ≤ ∑

|α|≤2m−1

‖bα‖∞‖Dαf‖

≤( ∑

|α|≤2m−1

‖bα‖∞)(

εθ−1‖P (D)f‖+ cε‖f‖).

It follows that D(P (D)) ⊂ D(Q(·, D)) and Q(·, D) is P (D)-bounded, and

since ε > 0 is arbitrary, the P (D)-bound of Q(·, D) is 0. Thus, by Remark 4.1.6,

it follows that P (D) +Q(·, D) generates a semigroup holomorphic in the right

half plane. �

7.2. The wave equation 141

Example 7.1.2. Let P (k) = −∑dj=1 k

2j and Q(x, k) =

∑dj=1 bj(x)kj , where

bj ∈ L∞(Rd). Then P,Q satisfy all the conditions of Theorem 7.1.1. Thus

A = P (D) +Q(·, D) generates a holomorphic semigroup of angle π/2, that is,

A =∑d

j=1 D2j +

∑dj=1 bj(x)Dj is the generator of a holomorphic semigroup of

angle π/2.

7.2 The wave equation

We consider the following initial value problem for the wave equation:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

∂2u

∂t2(x, t) = Δu(x, t), x ∈ Rd, t > 0,

u(x, 0) = f(x), x ∈ Rd,

∂u

∂t(x, 0) = g(x), x ∈ Rd.

(7.5)

We rewrite (7.5) as a first order problem in two dimensions, by setting

u1(x, t) = u(x, t) and u2(x, t) =∂u∂t (x, t), to get that⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

∂

∂t

(u1

u2

)=

(0 I

Δ 0

)(u1

u2

), x ∈ Rd, t > 0,

(u1(x, 0)

u2(x, 0)

)=

(f(x)

g(x)

), x ∈ Rd.

(7.6)

As mentioned earlier, we will write (7.6) as an abstract Cauchy problem

on an appropriate Banach space and show that the corresponding operator A is

the generator of a C0-semigroup. The right space in this instance is the Hilbert

space H = H1(Rd)⊕ L2(Rd) (see Appendix A.3).

Note that if we define, for (u1, u2) ∈ Y := C∞c (Rd) ⊕ C∞

c (Rd), the norm

‖ · ‖ by

‖(u1, u2)‖2 = ‖u1‖2H1(Rd) + ‖u2‖2L2(Rd),

then the completion of the normed space(Y, ‖ · ‖) is the Hilbert Space H.

We define the operator A associated with (7.6) as follows:

Definition 7.2.1. Let D(A) = H2(Rd)⊕H1(Rd) ⊂ H and define

A(u1, u2) = (u2,Δu1) for all (u1, u2) ∈ D(A).


Note that then the above problem may be written as the abstract Cauchy

problem u′(t) = Au(t), u(0) = u0 = (f, g) on the Hilbert space H.

Theorem 7.2.2. The operator A defined above is the infinitesimal generator of

a C0-semigroup {T (t)}t≥0 on H = H1(Rd)⊕ L2(Rd) satisfying

‖T (t)‖ ≤ e2t.

In fact, {T (t)}t≥0 extends to a C0-group on H.

Proof. We show first that for every f = (f1, f2) ∈ C∞c (Rd)⊕C∞

c (Rd) and real

λ �= 0, there is a unique element u = (u1, u2) ∈ D(A) such that (I − λA)u = f

and

‖u‖ ≤ (1− 2|λ|)−1‖f‖ for 0 < |λ| < 1

2.

Let f = (f1, f2) ∈ C∞c (Rd)⊕ C∞

c (Rd), and let k > 2. Set

wi(η) = (1 + λ2|η|2)−1fi(η).

Since fi ∈ C∞c (Rd) ⊂ H2(Rd), (1 + | · |2)k/2fi ∈ L2(Rd), by Lemma A.3.1 and

thus (1 + λ2|η|2)(k+2)/2wi(η) ∈ L2(Rd).

If we set

wi(x) = (2π)−d/2

∫Rd

ei〈x,η〉w(η) dη for i = 1, 2,

then wi ∈ Hk+2(Rd), again by Lemma A.3.1, and

wi − λ2Δwi = fi, i = 1, 2.

Setting u1 = w1 + λw2, u2 = w2 + λΔw1 and u =(u1, u2

)we see that

u ∈ Hk(Rd)⊕Hk−2(Rd) ⊂ D(A) = H2(Rd)⊕H1(Rd) for k ≥ 3, and

(I − λA)u =(u1 − λu2, u2 − λΔu1

)=(w1 − λ2Δw1, w2 − λ2Δw2

)=(f1, f2

)= f.

Observe that since fi ∈ C∞c (Rd), wi ∈ S(Rd), for i = 1, 2 and hence u1, u2 are

in the domain of Δ. We also note here that 〈u2,Δu1〉L2(Rd) = 〈Δu2, u1〉L2(Rd).

7.2. The wave equation 143

Moreover, since f1 = u1 − λu2 and f2 = u2 − λΔu1,

‖f‖2H = 〈f1, f1〉H1(Rd) + 〈f2, f2〉L2(Rd)

= 〈f1, f1〉L2(Rd) + 〈∇f1,∇f1〉L2(Rd) + 〈f2, f2〉L2(Rd)

= 〈f1, f1〉L2(Rd) − 〈Δf1, f1〉L2(Rd) + 〈f2, f2〉L2(Rd)

= 〈u1 − λu2 −Δu1 + λΔu2, u1 − λu2〉L2(Rd)

+ 〈u2 − λΔu1, u2 − λΔu1〉L2(Rd)

= 〈u1 −Δu1, u1〉L2(Rd) + ‖u2‖2L2(Rd) − 2λRe 〈u1, u2〉L2(Rd)

+ λ2‖Δu1‖2L2(Rd) + λ2‖u2‖L2(Rd)

− λ2〈Δu2, u2〉L2(Rd) − λ[〈u2,Δu1〉L2(Rd) − 〈Δu2, u1〉L2(Rd)

].

Since −Δ is a positive operator we conclude that

‖f‖2H ≥ 〈u1 −Δu1, u1〉L2(Rd) + ‖u2‖2L2(Rd) − 2|λ|Re 〈u1, u2〉L2(Rd)

≥ (1− |λ|)||u||2H.

Therefore, if λ ∈ R and 0 < |λ| < 12 , then

‖f‖2H ≥ (1 − 2|λ|)2‖u‖2H. (7.7)

Thus, whenever λ is real and 0 < |λ| < 12 , the range of (I − λA) contains

C∞c (Rd)⊕C∞

c (Rd), which is dense in H. But A is closed, and we have that the

range of (I − λA) is closed and hence it equals H. Therefore, for every f ∈ Hand real λ satisfying 0 < |λ| < 1

2 , there is a unique u ∈ D(A) such that

(I − λA)u = f and ‖u‖H ≥ (1− 2|λ|)−1‖f‖H.

This implies that (μ − A)−1 exists as a bounded operator on H, for real μ

satisfying |μ| > 2, and

‖(μ−A)−1‖ ≤ (|μ| − 2)−1 for such μ. (7.8)

Thus, (−∞,−2) ∪ (2,∞) ⊂ ρ(A) and ‖R(μ,A)‖ ≤ (|μ| − 2)−1 if |μ| > 2.

Noting that D(A) = H2(Rd) ⊕ H1(Rd) is dense in H, and A is closed, it

follows from Theorem 2.3.3 that A generates a C0-semigroup {T (t)}t≥0 on Hsatisfying ‖T (t)‖ ≤ e2t, for all t ≥ 0. Furthermore, since −A also generates a

C0-semigroup, {T (t)}t≥0 extends to a C0-group.

�


7.3 Schrodinger equation

In this section we consider the Schrodinger equation

1

i

∂u

∂t= Δu− V u, (7.9)

where the function V is called the potential. As before, we write (7.9) as an ab-

stract problem on a Hilbert space, L2(Rd), and use the theory of C0-semigroups

developed in the previous chapters. Let H = L2(Rd). Define the operators A

and B as follows:

Definition 7.3.1. Let D(A) = H2(Rd) and set for u ∈ D(A),

Au = iΔu = i

d∑j=1

∂2u

∂xj.

Definition 7.3.2. Let V be a measurable, finite almost everywhere defined func-

tion on Rd, with D(B) = {u ∈ L2(Rd), V u ∈ L2(Rd)} and set

(Bu)(x) = V (x)u(x) for all x ∈ Rd.

Theorem 7.3.3. The operator A defined above generates a C0-group of unitary

operators.

Proof. The result follows from Stone’s Theorem 3.2 if we can show that iA =

−Δ is selfadjoint. By the results on the Fourier transform in L2(Rn) in Lemma

A.2.3 and Remark A.2.6(d), it follows that iA is unitarily equivalent to the

operator of multiplication by the measurable non-negative function φ(k) = k2,

that is,

F(iAf) = Mφ(Ff)(k) for all f ∈ D(iA), where

D(iA) = {f ∈ L2(Rd :

∫ ∣∣φ(k)(Ff)(k)∣∣2 dk <∞}.

On the other hand, Theorem A.1.13 tells us that such an Mφ is selfadjoint (and

is also non-negative). �

We remark here that the operator −iA discussed in this section coincides

with the operator P (D) discussed in Section 7.1, with the polynomial P as

in Example 7.1.2. Hence the conclusion in Theorem 7.3.3 may alternatively

be deduced from the proof of Theorem 7.1.1, where it is shown that iP (D)

generates a unitary C0-group on R.

7.3. Schrodinger equation 145

Theorem 7.3.4. Let φ ∈ Lp(Rd) where p ≥ 2 and p > d/2. Then Mφ(iA+1)−1

is compact in L2(Rd).

Proof. Using results on the Fourier transform in Appendix (A.2) one con-

cludes that F(iA+1)−1F−1 = M(k2+1)−1 in L2(Rd). For any measurable func-

tion ψ : Rd → C, the operator Mφψ(iA) is an integral operator with kernel

(2π)−d/2φ(x)ψ(y − x) and therefore it follows easily that Mφψ(iA) is Hilbert-

Schmidt and hence compact if both φ and ψ ∈ L2(Rd). Indeed, for ψ ∈ Lp(Rd)

with p > d/2 and p ≥ 2, we have ψ(iA)f ∈ Lq(Rd), by the generalised Holder

inequality, where q−1 = p−1 + 1/2. Thus 1 ≤ q ≤ 2, and therefore, by the

Hausdorff-Young inequality (A.14) in Appendix A.2, the inverse Fourier trans-

form F−1(ψ(iA)f) ∈ Lr(Rd), with r−1 + q−1 = 1, such that

‖F−1(ψ(iA)f)‖r ≤ C(p)‖ψ(iA)f‖q.

From the two relations among p, q and r it follows that r−1+p−1 = 1/2 and an-

other application of the generalised Holder inequality yields that Mφψ(iA)f ∈L2(Rd), and that

‖Mφψ(iA)f‖2 ≤ ‖φ‖p∥∥F−1(ψ(iA)f )

∥∥r

≤ C(p)‖φ‖p‖ψ(iA)f‖q ≤ C(p)‖φ‖p‖ψ‖p‖f‖2.

This leads to the conclusion that Mφψ(iA) ∈ B(H) and

‖Mφψ(iA)‖ ≤ C(p)‖φ‖p‖ψ‖p. (7.10)

For φ, ψ ∈ Lp(Rd), set φm(x) = χm(x)φ(x), and ψm(iA) = χm(iA)ψ(A), where

m ∈ N, and χm(x) = χ{x:|x|≤m}(x) for all x ∈ Rd. Then φm and ψm ∈ L2(Rd)

for every m. By the earlier discussion, it follows that Mφmψm(iA) is Hilbert

Schmidt and hence compact in L2(Rd). By an application of the Dominated

Convergence Theorem one has that ‖φ−φm‖p and ‖ψ−ψm‖p → 0, as m→∞.

Therefore, by (7.10),

‖[Mφψ(iA)]− [Mφmψm(iA)]‖ ≤ ‖(Mφ −Mφm)ψ(iA)‖+ ‖Mφm(ψ(iA)− ψm(iA))‖

≤ C(p)(‖φ− φm‖p‖ψ‖p + ‖φm‖p‖ψ − ψm‖p

).

This estimate, together with the uniform boundedness of ‖φm‖p, implies that

Mφψ(iA) is the limit in operator norm of the sequence {Mφmψm(iA)}m of


Hilbert Schmidt operators and hence is compact. For the final conclusion it

remains only to convince ourselves that if ψ(k) = (k2 + 1)−1 for all k ∈ Rd,

then ψ ∈ Lp(Rd) with p > d/2 and p ≥ 2. �

Theorem 7.3.5. Suppose V (x) is real for all x ∈ Rd and V ∈ Lp(Rd), where

p > d/2 and p ≥ 2, and let B = MV , the operator of multiplication by V. Then

A − iB is the infinitesimal generator of a group of unitary operators L2(R2)

and σe(iA+B) = σe(iA).

Proof. By Theorem 7.3.4, MV (iA+1)−1 = B(iA+1)−1 is compact and there-

fore, by Theorem 4.2.1, B is iA compact. Therefore, by Theorem 4.2.6, iA+B

is selfadjoint and since iA = −Δ ≥ 0, iA + B is also bounded below, and

σe(iA + B) = σe(iA). The selfadjointness of iA + B together with Stone’s

Theorem 3.2 implies that A − iB = −i(iA + B) generates a group of unitary

operators. �

Appendix

We collect in this appendix concepts and results that are essential for the devel-

opment of the main theme of the book, though, in most cases, they constitute

basic materials. Therefore, very few proofs are included but relevant references

are given where required.

A.1 Unbounded operators

A linear operator A is bounded if it maps bounded sets into bounded sets

and this property is equivalent to saying that A is a continuous linear map,

that is, if xn → x ∈ X, then Axn → Ax. Also, this property allows A to

be defined, unambiguously, on whole of X, preserving the norm and hence,

one can assume without loss of generality that all such operators are defined

everywhere, constituting the Banach space B(X). However, a linear operator,

when not bounded, cannot be continuous and the property nearest to continuity

is that of closedness:

Definition A.1.1. Let X be a Banach space. A linear operator

A : D(A) ⊂ X → X is said to be

1. closed if for every Cauchy sequence {xn} ⊂ D(A) for which {Axn} is alsoa Cauchy sequence, one has x ≡ limn→∞ xn ∈ D(A) and limn Axn = Ax;

2. closable if {xn} ⊂ D(A) is such that xn → 0 and Axn → y, then y = 0.

Just as a bounded operator, a priori not defined everywhere, can be ex-

tended to an element of B(X), a linear operator A, if closable, can be extended

to a closed operator but there can be many closed extensions; the smallest of

these extensions (that is, the one with the smallest domain), called the closure

147© Springer Nature Singapore Pte Ltd. 2017 and Hindustan Book Agency 2017K.B. Sinha and S. Srivastava, Theory of Semigroups and Applications,Texts and Readings in Mathematics, DOI 10.1007/978-981-10-4864-7

148 Appendix

of A and denoted A, is obtained as follows:

D(A) := {x ∈ X : ∃ {xn} ⊂ D(A) and y ∈ X

such that {xn} → x and Axn → y},Ax = y for all x ∈ D(A).

One can check that the closability implies that A is a well defined operator with

D(A) ⊂ D(A), and also that A is closed if and only if A = A (Exercise A.1.2).

As an example, consider the (classical) differential operator in L2(R),

given by

A = −i ddx

with D(A) = C∞c (R),

the set of smooth functions on R with compact support, which is dense in

L2(R). It can be shown (Exercise A.1.3) that A is closable but not closed.

For this, one may note that the function f0(x) = exp(−|x|) is differentiable

everywhere except at x = 0 and f ′0 ∈ L2(R) but f0 does not belong in C∞

c (R).

We shall discuss this operator again in the next section on Fourier transforms.

Definition A.1.4. For a closed operator A on a Banach space X, the resolvent

set ρ(A) and the resolvent operator R(λ,A) of A are respectively defined as:

ρ(A) = {λ ∈ C : (λ−A)−1 ∈ B(X)} and R(λ,A) = (λ−A)−1.

Further, the spectrum of A shall be denoted by σ(A) := C \ ρ(A).

Then the following facts can be easily verified (Exercise A.1.5).

(1) For each λ, μ ∈ ρ(A), the following resolvent equation holds.

R(λ,A)− R(μ,A) = (μ− λ)R(λ,A)R(μ,A). (A.1)

(2) ρ(A) is open in C and the map ρ(A) � λ �→ R(λ,A) ∈ B(X) is norm-

holomorphic in each connected component of ρ(A). In particular, R(λ,A)

admits a power series expansion

R(λ,A) =

∞∑n=0

R(λ0, A)n+1(λ0−λ)n, for |λ−λ0| ≤ ‖R(λ0, A)‖−1. (A.2)

(3) σ(A) is a closed subset of C.

A.1. Unbounded operators 149

If the underlying space X is a Hilbert space, then the adjoint A∗ of a

densely defined linear operator A is defined as follows:

D(A∗) = {g ∈ H : ∃ g∗ ∈ H such that 〈g,Af〉 = 〈g∗, f〉 ∀f ∈ D(A)},A∗g = g∗ ∀g ∈ D(A∗).

For such an operator, one has the result that A is closable if and only if its ad-

joint A∗ is densely defined and in such a caseA = A∗∗ = (A∗)∗ (ExerciseA.1.6).

In a similar spirit, the adjoint A∗ for a linear, densely defined operator A

can be defined on a Banach space X :

D(A∗) = {g∗ ∈ X∗ : ∃h∗ ∈ X∗ such that 〈g∗, Af〉 = 〈h∗, f〉 ∀f ∈ D(A) ⊂ X},A∗g∗ = h∗ ∀g∗ ∈ D(A∗) ⊂ X∗.

An example of a linear (unbounded) operator which is not closable is the

following.

Example A.1.7. Let {ej}∞1 be an orthonormal basis in an infinite-dimensional

Hilbert space, let D(A) be the linear manifold of all finite linear combinations

of {ej}, and set Aej = je1 and extend linearly. It can be verified easily that

D(A∗) is not dense and A is not closable (Exercise A.1.8).

A dense subset S ⊂ D(A) is said to be a core for A if the closure A|S of

A restricted to S is exactly A. In the example of the differential operator A

in the paragraph preceding Definition A.1.4, C∞c (R) is a core for the closure

of A. A nice sufficient condition for a dense subset S ⊂ D(A) to be a core for

A is given in the following theorem, in the case when A is the generator of a

contraction semigroup {T (t)}t≥0.

Theorem A.1.9. Let {T (t)}t≥0 be a contraction C0-semigroup with generator

A in a Banach space X, and let S ⊂ D(A) be a dense subset in X such that

T (t)x ∈ S for every x ∈ S, for each t > 0. Then S is a core for A.

Proof. First we observe the following.

(i) For every y ∈ X, and t > 0, the strong integral

∫ t

0

T (s)y ds is defined as

the limit of the Riemann sum∑

P

[T (sj)y(sj+1 − sj)

] ≡ Ψ(y, P ; t) in X

as the width of the partition P ≡ {0 ≤ s1 < s2 < . . . < sn = t}, |P | =max(sj+1 − sj) → 0, and this limit exists in X since s �→ T (s)y ∈ X is

continuous (see Lemma 1.2.5).

150 Appendix

(ii) t−1

∫ t

0

T (s)y ds converges to y as t→ 0+, by Lemma 2.1.2.

(iii) By Lemma 1.2.5 and Theorem 2.2.7(c),

A

∫ t

0

T (s)y ds =

∫ t

0

AT (s)y ds = (T (t)− I)y,

for every y ∈ D(A).

Let x ∈ D(A), and let {xn} be a sequence of vectors in S such that xn → x and

Axn → u as n → ∞. Then by the hypothesis of the theorem, Ψ(xn, P ; t) ∈ S

for each n, and

limn→∞

lim|P |→0

t−1Ψ(xn, P ; t) = t−1

∫ t

0

T (s)x ds, (A.3)

while

limn→∞

lim|P |→0

t−1AΨ(xn, P ; t) = limn→∞

lim|P |→0

t−1Ψ(Axn, P ; t) (A.4)

= t−1(T (t)− I)x,

for every t > 0. On the other hand, since Axn → u as n→∞, it follows that

limn→∞

lim|P |→0

t−1Ψ(Axn, P ; t) = t−1

∫ t

0

T (s)u ds

for every t > 0. Therefore,

t−1

∫ t

0

T (s)u ds = t−1(T (t)− I)x = t−1

∫ t

0

T (s)Axds.

Taking the limit t→ 0+ in the above equality we have using (ii) that u = Ax.

This proves that S is a core for A. �

LetH be a Hilbert space. A densely defined operator A is said to be a sym-

metric operator if A ⊂ A∗, that is, D(A) ⊂ D(A∗) and Af = A∗f for all f ∈D(A), and it is said to be selfadjoint if furthermore D(A) = D(A∗). While ev-

ery bounded symmetric operator with D(A) = H is selfadjoint, the condition

A = A∗ puts severe restrictions in the unbounded case. A symmetric operator

may have many closed symmetric extensions, including selfadjoint ones, as the

following example shows.

A.1. Unbounded operators 151

Example A.1.10. Let H = L2[0, 1], and set

D(A0) = {f ∈ H : f absolutely continuous, f ′ ∈ L2, f(0) = f(1) = 0},(A0f)x = −if ′(x).

One can compute the adjoint A∗0 to be

D(A∗0) = {f ∈ H : f absolutely continuous, f ′ ∈ L2}

(A∗0f)x = −if ′(x),

showing that A0 is symmetric but not selfadjoint, that is, A0 � A∗0. One can

compute all the selfadjoint extensions of A0 (Exercise A.1.11) to find that there

are uncountably many selfadjoint extensions, denoted by Aα, for α ∈ C, |α| = 1,

with

D(Aα) = {f ∈ H : f absolutely continuous, f ′ ∈ L2, f(1) = αf(0)}.

A criterion for selfadjointness is included in the next result:

Theorem A.1.12. (i) For a symmetric operator A, x ∈ D(A) and z ∈ C, with

Im z �= 0,

‖(A+ z)x‖2 = ‖(A+ Re z)x‖2 + |Im z|2‖x‖2.

(ii) A closed symmetric operator is selfadjoint if and only if the ranges of the

operators A± i are both H.

From Definition A.1.4 and Theorem A.1.12, it can be shown that (see for

example [1, pages 56 and 200]) the open upper and lower half planes in C are

contained in the resolvent set of a selfadjoint operator in a Hilbert space and

hence its spectrum is a closed subset of R.

One of the simplest, in fact, almost canonical examples of a selfadjoint

operator in a Hilbert space L2(μ) is that of an operator of multiplication by a

real-valued, measurable function φ : Ω→ C with

D(Mφ) = {f ∈ L2(μ) : φf ∈ L2(μ)}(Mφf)(x) = φ(x)f(x) ∀f ∈ D(Mφ), and x ∈ Ω.

In fact, the above definition extends to Lp(μ), 1 ≤ p <∞, with

D(Mφ) = {f ∈ Lp(μ) : φf ∈ Lp(μ)}.

152 Appendix

Theorem A.1.13. Let 1 ≤ p <∞ and φ be as above. Then

(i) Mφ in Lp(μ) is bounded if and only if φ ∈ L∞ and ‖Mφ‖ = ‖φ‖∞;

(ii) Mφ in Lp(μ) is a closed, densely defined operator and

σ(Mφ) = essential range of φ

≡⋂ε>0

{λ ∈ C : μ{x ∈ Ω : |φ(x) − λ| < ε} > 0

};

(iii) if p = 2 and if φ is real valued, then Mφ is selfadjoint in L2(μ).

Proof. (i) It is clear that Mφ is bounded if φ ∈ L∞(μ), and in such a case,

‖Mφ‖ ≤ ‖φ‖∞. The equality follows from the definition of L∞ norm as

essential supremum (Exercise A.1.14). Conversely, if φ is not essentially

bounded, then for every m ∈ N, if we set Sm = {x ∈ Ω : |φ(x)| ≥ m}, Sm

will have a non-zero positive measure and therefore ‖Mφf‖p ≥ m‖f‖p, iff = χSm �= 0.

(ii) Let f ∈ Lp(μ). The sequence {fm}, given by

fm(x) =(1 +

|φ(x)|m

)−1

f(x), m = 1, 2, . . . ,

is well defined, measurable and satisfies |fm(x)| ≤ |f(x)| ∀x ∈ Ω and

fm(x) → f(x) pointwise as m → ∞. An application of the Dominated

Convergence Theorem proves the convergence of fm to f in Lp(μ). It is

clear that for each m, fm ∈ D(Mφ), proving that Mφ is densely defined.

That Mφ is closed is a simple consequence of the fact that if gm → g,

in Lp(μ) norm, as m → ∞, then gmk(x) → g(x) pointwise μ-almost

everywhere, as k →∞, for a subsequence {mk} ⊂ {m}.

Now suppose λ /∈ ess ranφ. Then there exists δ > 0 such that

|φ(x) − λ| > δ for μ− almost all x.

Thus ψ(x) = (φ(x)−λ)−1 is in L∞(μ) and by (i) above, Mψ is a bounded

inverse of Mφ − λ. Therefore, λ ∈ ρ(Mφ). Conversely, if λ ∈ ρ(Mφ),

then M(φ−λ)−1 is a bounded operator, which on using (i) shows that λ /∈ess ranφ.

A.2. Fourier transforms 153

(iii) If φ is real valued, then it follows from the definition thatMφ is symmetric

in the Hilbert space L2(μ) and is closed. Furthermore, φ(x) ± i �= 0, and

|(φ(x) ± i)−1| ≤ 1 ∀x ∈ Ω. Therefore every f ∈ L2(μ) may be written

as f(x) =((Mφ ± i)g

)(x) for g(x) = (φ(x) ± i)−1f(x) ∈ L2(μ), and this

shows that Ran(Mφ ± i) = L2(μ). Hence self-adjointness follows from

Theorem A.1.12(ii).

�

We shall often need the following form of the Spectral Theorem.

Theorem A.1.15. Each selfadjoint operator is unitarily equivalent to a real mul-

tiplication operator. More precisely, if A is a selfadjoint operator on a Hilbert

space H, then there exist a σ-finite measure space (Ω,�, μ) and a measurable

function φ : Ω → R, such that A is unitarily equivalent to the multiplication

operator Mφ on L2(μ).

We refer to [1], [20] and [21] for proofs and discussions on the material

presented in Appendix A.1 as well as in Appendix A.2.

A.2 Fourier transforms

Here we shall define and discuss the Fourier transform, which has very wide

applications in various areas of analysis.

By a multi-index α we shall mean that α = (α1, α2, . . . , αd) where

α1, . . . , αd ∈ N⋃{0}. Define |α| := ∑j=d

j=1 αj , and xα = xα11 xα2

2 · · ·xαd

d , for

all x ∈ Rd and Dα = Dα11 · · ·Dαd

d , where Dj =∂∂xj

. Thus,

Dα = Dα11 . . .Dαd

d =∂α1

∂xα11

∂α2

∂xα22

· · · ∂αd

∂xαd

d

.

First we define the class S(Rd), called the Schwartz class of functions on

Rd, the linear space of arbitrarily often differentiable complex-valued functions

of rapid decrease at infinity as:

S(Rd) ≡ {f ∈ C∞(Rd) : cα,l(f) <∞ for every l ∈ N ∪ {0} and multi-index α

}where

cα,l(f) := supx∈Rd

∣∣(1 + |x|2)l/2(Dαf)(x)∣∣. (A.5)

154 Appendix

A typical member of S(Rd) is the Gaussian function h0(x) = exp(− 1

2 |x|2)

and also note that C∞c (Rd) is a strict subset of S(Rd) and therefore S(Rd) is

dense in L2(Rd), in fact in every Lp(Rd) for 1 � p <∞. For f ∈ L1(Rd), define

the Fourier transform f of f as:

f(k) = (2π)−d/2

∫f(x)e−i〈k,x〉 dx, (A.6)

where k ∈ Rd and 〈., .〉 is the Euclidean inner product on Rd. It is clear that

f is well defined everywhere in Rd as a bounded continuous function which

converges to 0 at∞ and hence is uniformly continuous in Rd. (Exercise A.2.1).

It is useful to compute the Fourier transform of h0 to find that

h0(k) = exp(− 1

2|k|2

)= h0(k). (A.7)

The details of this computation are left as (Exercise A.2.2).

Lemma A.2.3. Let f ∈ S(Rd). Then

(i)(

∂lf∂xj

l

)(k)=(ikj

)lf(k) ∀ k ∈ Rd, j ∈ {1, 2, . . . , d}, l ∈ N;

(ii)(xj

lf)(k)= il

(∂lf∂kj

l

)(k) ∀ k ∈ Rd, j ∈ {1, 2, . . . , n}, l ∈ N.

The proof consists of treating the defining integral (A.6) for(

∂lf∂xj

l

)(k)

and(xj

lf)(k)respectively as an iterated integral, performing an integration

by parts in the jth integral and using the property (A.5) of rapid decrease at in-

finity of f and all its partial derivatives. The details are left as (Exercise A.2.4).

The next theorem is an immediate consequence of this lemma and sums

up the essential properties of the Fourier transform map in the Hilbert space

L2(Rd).

Theorem A.2.5. (i) S(Rd) is invariant under the Fourier transformation and

the map defined as Ff(k) = f(k) ∀ k ∈ Rd and f ∈ S(Rd) maps S(Rd)

bijectively to S(Rd).

(ii) (Plancherel) Furthermore, F is an isometry in L2(Rd)-norm on S(Rd) and

thus extends to a unitary operator on L2(Rd). (We denote the unitary

extension by the same symbol F).Proof. (i) By virtue of Lemma A.2.3 and the definition of the Schwartz class

(A.5), it follows easily that f ∈ C∞(Rd) for every f ∈ S(Rd). Combining both


the results of Lemma A.2.3, we get that, for any l ∈ 2N,

(1 + |k|2)l/2( ∂|m|f

∂km11 ... ∂kmd

d

)(k) = F{(1−Δ)l/2

[(−ix1)

m1 ...(−ixd)mdf

]}(k),

proving that f ∈ S(Rd).

For the second part of (i), we need to construct the inverse Fourier trans-

form. Let f , g ∈ S(Rd), α > 0 and y ∈ Rd. Then∫Rd

g(αk)f(k)ei〈k,y〉dk = (2π)−d/2

∫g(αk)

∫f(x)ei〈k,y−x〉 dx dk

= α−d

∫f(x)

{(2π)−d/2

∫g(k)ei〈

y−xα ,k〉 dk

}dx (A.8)

where we have used Fubini’s theorem to change the order of integration and

made a change of variable k −→ αk ∈ Rd. Continuing with the above compu-

tation, we get that the right hand side of (A.8) equals

α−d

∫f(x)g

(x− y

α

)dx =

∫f(αu+ y)g(u) du. (A.9)

Letting α −→ 0+, by the Dominated Convergence Theorem and the properties

of the function f and g, the right hand side of (A.9) converges to f(y)

∫g(u) du.

Thus, letting α→ 0+ in (A.8) gives

g(0)

∫f(k)ei〈k,y〉dk = f(y)

∫g(u) du. (A.10)

Next we pick g to be the special function h0 and note from (A.7) that∫h0(u)du =

∫h0(u)du =

∫e−|u|2/2 du = (2π)d/2

to lead to the inverse Fourier transform:

f(y) = (2π)−d/2

∫f(k)ei〈k,y〉 dk. (A.11)

This completes the proof of (i).

(ii) Using the symbol F for the map of the Fourier transform in equation

(A.8) and setting α = 1 and y = 0 ∈ Rd, we get∫g(k)(Ff)(k) dk =

∫f(u)(Fg)(u) du. (A.12)

156 Appendix

Now set h = Fg (which belongs to S(Rd) by part (i)) and observe that

g(k) = (F−1h)(k) = (2π)−d/2

∫h(u)ei〈k,u〉 du

={(2π)−d/2

∫h(u)e−i〈k,u〉du

}=

¯h(k).

Substituting the above in (A.12) leads to the Plancherel relation, viz.

⟨Ff,Fh⟩L2(Rd)

=

∫f(k)h(k) dk =

⟨f, h

⟩L2(Rd)

. (A.13)

Thus, F defined on S(Rd) is an isometric map with range equal to S(Rd), which

is dense in L2(Rd), and therefore, by standard arguments, extends to a unitary

operator in L2(Rd). �

Remark A.2.6. (a) Note that, though the expression for f ∈ S(Rd),

f(k) ≡ Ff(k) = (2π)−d/2

∫f(x)e−i〈k,x〉 dx,

makes sense for every k ∈ Rd, the extension, viz. Ff(k) for f ∈ L2(Rd)

makes sense for almost all (Lebesgue) k ∈ Rd and the right hand side of

the above identity has to be interpreted as such. However, the sequence

of characteristic functions {χn} of balls of radius N about the origin in

L2(Rd) converges strongly to the identity operator in L2(Rd) and we have

that ∥∥Ff −F(χNf)∥∥22=∥∥f − χNf

∥∥22−→ 0

as N −→∞. Hence as N −→∞, (2π)−d/2

∫χNf(x)e−i〈k,x〉dx converges

to Ff(k) in the L2-sense. This in turn implies that for a suitable subse-

quence {Nl} ⊂ {N}, (2π)−d/2

∫|x|≤Nl

f(x)e−i〈k,x〉 dx converges pointwise

almost everywhere to (Ff)(k).

(b) Consider Rd as the locally compact abelian group under addition (called

the translation group in Rd) with the Lebesgue measure as the Haar (in-

variant) measure on that group. Denote the associated transformation Ty

(for y ∈ Rd) as (Tyf

)(x) := f(x− y).


Then it is clear that for f, g ∈ L2(Rd), Tyf ∈ L2(Rd),⟨Tyf, Tyg

⟩=⟨f, g

⟩and (

Tyf)(k) = (2π)−d/2

∫Tyf(x)e

−i〈k,x〉dx

= (2π)−d/2

∫f(x− y)e−i〈k,x−y+y〉dx

= e−i〈k,y〉f(k),

that is, the factor exp(−i

⟨k, .

⟩)is the (one-dimensional irreducible repre-

sentation) character of the translation group Rd in its regular representa-

tion in L2(Rd). This result can easily be extended to any Lp(Rd) (except

for the Plancherel relation) whenever translation can be defined.

(c) So far we have only talked about the Fourier transform on L2(Rd) and

the question arises about how much of these results can be extended to

Lp with arbitrary p ∈ [1,∞). By what we have said so far, it is clear that

the Fourier transform map F has the following properties.

(i) F : L1(Rd) −→ L∞(Rd) and ‖Ff‖∞ � (2π)−d/2‖f‖1;(ii) F : L2(Rd) −→ L2(Rd) and ‖Ff‖2 = ‖f‖2.Therefore, interpolating (see pages 11–34 of [21]) between p = 1 and p = 2,

one gets that

F : Lp(Rd) −→ Lq(Rd) and ‖Ff‖q � (2π)d(

12−p−1

)‖f‖p, (A.14)

where 1 � p � 2 and q−1 + p−1 = 1 (called the Hausdorff-Young inequal-

ity). For 2 < p <∞, in general, the Fourier transform of f ∈ Lp(Rd) need

not be a function, defined almost everywhere. However, a wider meaning

can be associated to the Fourier transform, making it meaningful, in the

sense of distributions (see, for example, [22]).

(d) The results of Lemma A.2.3 can be used to give meaning to the operation

of differentiation, well beyond the usual classical meaning of it. In other

words, if we rewrite the result(F( ∂

∂xjf))

(k) = ikj(Ff)(k) as F ∂

∂xj=(ikj

)F , for each j = 1, 2 . . . d,

then this gives rise to the following.

D( ∂

∂xj

)={f ∈ L2(Rd) :

∫ ∣∣kj f(k)∣∣2dk <∞}

158 Appendix

and the action is given as above. From this one can show, for example,

that if f ∈ D( ∂∂xj

)(j = 1, 2, ..., d), then f is bounded and continuous, but

in general need not be classically differentiable (Exercise A.2.7).

A.3 Sobolev spaces

In this section we define some Sobolev spaces and some results concerning them

that we need. Although the ideas are implicit in our treatment, we avoid using

explicitly the language of distributions. For details and proofs of the results

quoted here we refer the reader to [3] and [22].

Let Ω = Rd for d > 1, or let Ω be an open interval of R, finite or infinite.

Recall that C∞c (Ω) is the space of infinitely differentiable functions with com-

pact support in Ω (also called the space of test functions on Ω) while Lp(Ω)

denotes, as usual, the Lebesgue measurable functions on Ω whose pth power is

Lebesgue integrable. For k ∈ N and 1 ≤ p ≤ ∞, the Sobolev space W k,p(Ω) is

defined as

W k,p(Ω) =

{f ∈ Lp(Ω) : for each multi-index α with

|α| ≤ k, there exists fα ∈ Lp(Ω) such that∫Ω

fαφdx = (−1)|α|∫Ω

f(Dαφ

)dx ∀φ ∈ C∞

c (Ω)

},

where Dα has been defined earlier in Section A.2. The element fα in the above

definition is called the weak derivative of f of order α and we write Dαf := fα.

Thus, W k,p(Ω) consists of those functions f in Lp(Ω) whose weak deriva-

tive Dαf lies in Lp(Ω) for every multi-index α with |α| ≤ k; clearly, W 0,p(Ω) =

Lp(Ω).

W k,p(Ω) is a Banach space with respect to the norm

‖f‖k,p ≡ ‖f‖Wk,p(Ω) :=∑|α|≤k

‖Dαf‖p

Further, we denote by W k,p0 (Ω) the closure of C∞

c (Ω) in W k,p(Ω).

For p = 2, the following defines an equivalent norm on W 2,p(Ω):

‖f‖k,2 :=( ∑

|α|≤k

‖Dαf‖22)1/2

.

A.3. Sobolev spaces 159

This makes W 2,p(Ω) a Hilbert space with the inner product

〈f, g〉 =∑|α|≤k

∫Ω

DαfDαg dx.

We write

Hk(Ω) := W k,2(Ω) and Hk0 := W k,2

0 (Ω),

as has been noted earlier in Chapter 7.

The following characterisation in terms of Fourier transforms holds for

the spaces Hk(Rd), as has been noted earlier in Remark A.2.6(d).

Lemma A.3.1. Let k ∈ N ∪ {0} and f ∈ L2(Rd). Then f ∈ Hk(Rd) if and only

if the function x �→ (1 + |x|2)k/2Ff(x) ∈ L2(Rd).

When d = 1 and Ω is an interval in R (bounded or unbounded) we have

the following useful characterisation of W k,p(Ω).

Lemma A.3.2. Let 1 ≤ p <∞ and Ω be an interval in R. Then f ∈W k,p(Ω) if

and only if f has continuous derivatives up to order (k−1), f (k−1) is absolutely

continuous and f (j) ∈ Lp(Ω) for j ∈ {0, 1, . . . , k}.Proof. We include a brief sketch of a proof. From the definition of W k,p(Ω), it

follows that for k ≥ 2,

W k,p(Ω) = {f ∈W (k−1),p(Ω) : Df ∈W (k−1),p(Ω)}.Thus, it can be seen that if the statement of the above theorem holds for

k ∈ N, then it holds for k + 1 as well. So it is enough to prove the result for

k = 1. Now if f is absolutely continuous, and f ′ ∈ Lp(Ω), then it is clear that

f ∈ W 1,p(Ω). Conversely, f ∈ W 1,p(Ω) implies that f ′ ∈ Lp(Ω). Therefore, by

Holder’s inequality, it follows that f ′ is in L1loc(Ω). The fundamental theorem

of integral calculus now implies that f is absolutely continuous. �

Finally, for a function f ∈ W 2,p(Ω) where 1 ≤ p < ∞, we define the

Laplacian Δf of f as

Δf =

d∑i=1

∂2f

∂xj2. (A.15)

For f, g ∈ H2(Ω), using Lemma A.2.3 and Plancherel’s relation (Theorem

A.2.5(ii)) we have:

〈Δf, g〉L2(Ω) = 〈F(Δf),Fg〉L2 = −〈F∇f,F∇g〉L2 = −〈∇f,∇g〉L2(Ω),

(A.16)

160 Appendix

where

∇f = gradient f =( ∂f

∂x1,∂f

∂x2, . . . ,

∂f

∂xd

).

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Index

A∗, 36, 149

A, 148

BUC(R+), 19

BUC1(R+), 38

B(X), 1

C1[0, 1], 60

C[0, 1], 60

C∞c (R+), 39

C0(R+), 16

C0, 19

D(·) , 22Δ, 47, 159

D, 138

Dα, 153

E, 106, 116

Ff , 64, 154H1(Rd), 141

H2(Rd), 63

Hk(Ω), 159

l1R, 116

l∞R, 116

Lp[a, b], 42

Lp(R+), 39

Lp(R), 41

L1loc(R), 48

L1R(Ω,�, μ), 116

L∞R(Ω,�, μ), 116

Mφ, 42, 77, 151

Mn(C), 21

R(λ,A), 148

R+, 5

ρ(A), 26, 148

Sθ, 67

S(Rd), 153

σ(A), 26, 148

σd(A), 88

σe(A), 88

T , 13

T ∗, 36

T∗, 117

W 1,p(R+), 39

W k,p(Ω), 158

X∗, 2

X∗, 116

X , 1

χ�, 1, 2, 5

almost separably valued, 2

Banach algebra, 10

birth process, 135

Bochner integrable, 5

Bochner integral, 6


166 Index

Brownian motion, 44, 105

standard, 105

Cauchy measure, 47

Central Limit Theorem, 108, 111

Chapman-Kolmogorov equation,

119

Chernoff’s Theorem, 97

convergence of semigroups, 90

death process, 134

determining set, 3

differential operator, 138

discrete state space, 118

dual injection, 59

ergodic, 111

essential range, 42

expectation, 106, 116

Feynman-Kac Formula, 102

finitely valued, 2

Fourier transform, 154

Gaussian function, 154

generator

heat semigroup, 47

holomorphic semigroup, 67

multiplication semigroup, 43

perturbation of, 81

relatively bounded

perturbation of, 84

group

C0, 31

generator, 31

isometry, 41

Schrodinger free evolution, 79

unitary, 64

Hausdorff-Young inequality, 157

Hille Yosida Theorem, 32, 35

holomorphic function, 66

weakly, 66

strongly, 66

Jensen’s inequality, 45, 49

Laplacian, 47, 159

Lie Product Formula, 100

Lumer Phillips Theorem, 61

Markov maps, 116

Markov semigroup, 117

Mean Ergodic Theorem, 111, 112

multi-index, 153

normal probability measure, 47

operator

adjoint of, 149

accretive, 53

closable, 147

closed, 147

closure of, 148

core of, 98, 149

dissipative, 53, 59

maximal dissipative, 53

multiplication, 152

relatively bounded, 84, 86

relatively compact, 86

resolvent of, 148

sectorial, 69

selfadjoint, 78, 150

symmetric, 150

Index 167

unbounded, 147

operator valued function, 12

strongly continuous, 13

strongly measurable, 13

uniformly continuous, 13

uniformly measurable, 12

weakly continuous, 13

weakly measurable, 13

Plancherel Relation, 156

positive cone, 116, 117

relative bound, 84

resolvent, 26

resolvent equation, 148

resolvent set, 26

Schrodinger equation, 144

Schwartz class, 153

semigroup, 13

C0, 19, 25

adjoint, 36

bounded holomorphic, 67

contraction, 19

convolution, 44

diffusion, 44

exponential growth bound, 17

Gaussian, 44

generator, 26

heat, 44, 47

holomorphic, 66, 67

left shift, 41

minimal, 130

multiplication, 42, 43, 77

nilpotent, 42

right shift, 41

strongly continuous, 19

translation, 41

type, 17

separably valued, 2

simple function, 2

Sobolev space, 158

Spectral Theorem, 78, 153

spectrum, 26

discrete, 88

essential, 88

of selfadjoint operator, 151

Stone’s Theorem, 64

strictly elliptic, 138

strongly measurable, 2

subadditive function, 10

test functions, 158

Trotter-Kato Product, 99

wave equation, 141

weak derivative , 158

weakly elliptic, 138

weakly measurable, 2

Wiener measure, 103, 105

Texts and Readings in Mathematics

1. R. B. Bapat: Linear Algebra and Linear Models (Third Edition)

2. Rajendra Bhatia: Fourier Series (Second Edition)

3. C. Musili: Representations of Finite Groups

4. H. Helson: Linear Algebra (Second Edition)

5. D. Sarason: Complex Function Theory (Second Edition)

6. M. G. Nadkarni: Basic Ergodic Theory (Third Edition)

7. H. Helson: Harmonic Analysis (Second Edition)

8. K. Chandrasekharan: A Course on Integration Theory

9. K. Chandrasekharan: A Course on Topological Groups

10. R. Bhatia (ed.): Analysis, Geometry and Probability

11. K. R. Davidson: C*-Algebras by Example (Reprint)

12. M. Bhattacharjee et al.: Notes on Infinite Permutation Groups

13. V. S. Sunder: Functional Analysis – Spectral Theory

14 . V. S. Varadarajan: Algebra in Ancient and Modern Times

15. M. G. Nadkarni: Spectral Theory of Dynamical Systems

16. A. Borel: Semisimple Groups and Riemannian Symmetric Spaces

17. M. Marcolli: Seiberg-Witten Gauge Theory

18. A. Bottcher and S. M. Grudsky: Toeplitz Matrices, Asymptotic Linear

Algebra and Functional Analysis

19. A. R. Rao and P. Bhimasankaram: Linear Algebra (Second Edition)

20. C. Musili: Algebraic Geometry for Beginners

21. A. R. Rajwade: Convex Polyhedra with Regularity Conditions and Hilbert's

Third Problem

22. S. Kumaresan: A Course in Differential Geometry and Lie Groups

23. Stef Tijs: Introduction to Game Theory

24. B. Sury: The Congruence Subgroup Problem

25. R. Bhatia (ed.): Connected at Infinity

26. K. Mukherjea: Differential Calculus in Normed Linear Spaces (Second

Edition)

27. Satya Deo: Algebraic Topology: A Primer (Corrected Reprint)

28. S. Kesavan: Nonlinear Functional Analysis: A First Course

29. S. Szabó: Topics in Factorization of Abelian Groups

30. S. Kumaresan and G. Santhanam: An Expedition to Geometry

31. D. Mumford: Lectures on Curves on an Algebraic Surface (Reprint)

32. J. W. Milnor and J. D. Stasheff: Characteristic Classes (Reprint)

33. K. R. Parthasarathy: Introduction to Probability and Measure

(Corrected Reprint)

34. Amiya Mukherjee: Topics in Differential Topology

35. K. R. Parthasarathy: Mathematical Foundations of Quantum Mechanics

36. K. B. Athreya and S. N. Lahiri: Measure Theory

37. Terence Tao: Analysis I (Third Edition)

38. Terence Tao: Analysis II (Third Edition)

39. W. Decker and C. Lossen: Computing in Algebraic Geometry

40. A. Goswami and B. V. Rao: A Course in Applied Stochastic Processes

41. K. B. Athreya and S. N. Lahiri: Probability Theory

42. A. R. Rajwade and A. K. Bhandari: Surprises and Counterexamples in

Real Function Theory

43. G. H. Golub and C. F. Van Loan: Matrix Computations (Reprint of the

Fourth Edition)

44. Rajendra Bhatia: Positive Definite Matrices

45. K. R. Parthasarathy: Coding Theorems of Classical and Quantum

Information Theory (Second Edition)

46. C. S. Seshadri: Introduction to the Theory of Standard Monomials

(Second Edition)

47. Alain Connes and Matilde Marcolli: Noncommutative Geometry, Quantum

Fields and Motives

48. Vivek S. Borkar: Stochastic Approximation: A Dynamical Systems

Viewpoint

49. B. J. Venkatachala: Inequalities: An Approach Through Problems

50. Rajendra Bhatia: Notes on Functional Analysis

51. A. Clebsch (ed.): Jacobi's Lectures on Dynamics (Second Revised Edition)

52. S. Kesavan: Functional Analysis

53. V. Lakshmibai and Justin Brown: Flag Varieties: An Interplay of Geometry,

Combinatorics, and Representation Theory

54. S. Ramasubramanian: Lectures on Insurance Models

55. Sebastian M. Cioaba and M. Ram Murty: A First Course in Graph Theory

and Combinatorics

56. Bamdad R. Yahaghi: Iranian Mathematics Competitions, 1973-2007

57. Aloke Dey: Incomplete Block Designs

58. R. B. Bapat: Graphs and Matrices (Second Edition)

59. Hermann Weyl: Algebraic Theory of Numbers (Reprint)

60. Carl Ludwig Siegel: Transcendental Numbers (Reprint)

61. Steven J. Miller and Ramin Takloo-Bighash: An Invitation to Number Theory

(Reprint)

62. John Milnor: Dynamics in One Complex Variable (Reprint)

63. R. P. Pakshirajan: Probability Theory: A Foundational Course

64. Sharad S. Sane: Combinatorial Techniques

65. Hermann Weyl: The Classical Groups: Their Invariants and Representations

(Reprint)

66. John Milnor: Morse Theory (Reprint)

67. R. Bhatia (ed.): Connected at Infinity II

68. Donald Passman: A Course in Ring Theory (Reprint)

69. Amiya Mukherjee: Atiyah-Singer Index Theorem: An Introduction

70. Fumio Hiai and Dénes Petz: Introduction to Matrix Analysis and

Applications

71. V. S. Sunder: Operators on Hilbert Spaces

72. Amiya Mukherjee: Differential Topology

73. David Mumford and Tadao Oda: Algebraic Geometry II

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