operator stieltjes integrals with respect to a spectral ... · in the usual way as the limits (if...

Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc.Tom 72 (2011), vyp. 1 2011, Pages 45–77

S 0077-1554(2012)00195-2Article electronically published on January 12, 2012

OPERATOR STIELTJES INTEGRALS

WITH RESPECT TO A SPECTRAL MEASURE

AND SOLUTIONS OF SOME OPERATOR EQUATIONS

S. ALBEVERIO AND A. K. MOTOVILOV

Abstract. We introduce the notion of Stieltjes integral with respect to the spectralmeasure corresponding to a normal operator. Sufficient conditions for the existenceof this integral are given, and estimates for its norm are established. The results areapplied to operator Sylvester and Riccati equations. Assuming that the spectrum of aclosed densely defined operator A does not have common points with the spectrum ofa normal operator C and that D is a bounded operator, we construct a representationof a strong solution X of the Sylvester equation XA − CX = D in the form of anoperator Stieltjes integral with respect to the spectral measure of C. On the basis ofthis result, we establish sufficient conditions for the existence of a strong solution ofthe operator Riccati equation Y A − CY + Y BY = D, where B is another boundedoperator.

1. Introduction

We consider integrals that can formally be written as

(1.1)

∫Ω

F (z)dE(z) or

∫Ω

dE(z)G(z),

where Ω is a Borel subset of the complex plane C and E(·) is the spectral measurecorresponding to a normal operator on a Hilbert space K. The operator-valued functionF is assumed to be defined on Ω and to range in the space B(K,H) of bounded operatorsfrom K to a Hilbert space H. Likewise, G is assumed to be an operator-valued functionthat maps Ω into B(H,K). Clearly, any reasonable definition of integration in (1.1) shouldresult in operators from K to H and from H to K, respectively.

Integrals of the form (1.1) are already of interest in themselves. But they also arise innumerous applications, in particular, when studying perturbations of spectral subspaces(e.g., see [1]). Integrals of this kind prove quite useful when studying operator Sylvesterand Riccati equations (see [2, 4, 23, 26]).

There is an important special case in which the function F (z) admits the productrepresentation

(1.2) F (z) = ϕ(A, z)T,

where ϕ(ζ, z) is a “sufficiently good” scalar function of the complex variables ζ and z, Ais a normal operator on H, and T is a bounded operator from K to H. In this case, the

2010 Mathematics Subject Classification. Primary 47B15; Secondary 47A56, 47A62.Key words and phrases. Operator Stieltjes integral, operator-valued function, normal operator, spec-

tral measure, Sylvester equation, Riccati equation.Supported by the Russian Foundation for Basic Research, Deutsche Forschungsgemeinschaft, and the

Heisenberg–Landau Program.A. K. Motovilov is grateful to the Institute for Applied Mathematics, University of Bonn, for kind

hospitality when carrying out this research.

c©2012 American Mathematical Society

45

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

46 S. ALBEVERIO AND A. K. MOTOVILOV

integral∫ΩF (z)dE(z) can be treated as a double operator Stieltjes integral [8]. More

precisely,

(1.3)

∫Ω

F (z)dE(z) =

∫˜Ω

∫Ω

ϕ(ζ, z) dE(ζ)T dE(z),

where E(·) is the spectral measure associated with A and Ω = spec(A). By now, thereexists a virtually comprehensive theory of double operator Stieltjes integrals, which ismainly due to Birman and Solomyak [9, 10, 11]. (See also the papers mentioned in [8].)

The situation is completely different for more general integrals of the form (1.1) wherethe operator-valued functions F (z) and G(z) do not admit a representation similar to(1.2) in terms of scalar functions of normal operators. To the best of the authors’knowledge, there is a sufficiently well-developed approach to operator integrals of theform (1.1) only for the case in which the spectral measure E(·) corresponds to a self-adjoint operator. (See [1, 3, 4, 23] and the papers cited therein.) In particular, it wasproved in [1] that the integrals (1.1) exist as Riemann–Stieltjes integrals if Ω is a finiteinterval of the real line and the operator-valued functions F and G are Lipschitz onΩ. Moreover, in this case the integrals (1.1) exist in the sense of the uniform operatortopology. Some sufficient conditions for the existence of improper integrals of the typeof (1.1) for the case in which the spectral measure E(·) corresponds to an unboundedself-adjoint operator can be found in [4, 23].

The aim of the present paper is to extend the main notions and results of the theoryof operator Stieltjes integrals to the case in which the spectral measure E(·) correspondsto an arbitrary normal operator. In fact, we consider the integrals (1.1) in a somewhatmore general setting that may involve functions F (λ, μ) and G(λ, μ) of two real variablesλ = Re z and μ = Im z, that is, functions that may depend not only on the complexvariable z but also on the complex conjugate variable z. The integrals (1.1) are introducedin the usual way as the limits (if these exist) of the corresponding Riemann–Stieltjesintegral sums as the rank of the partition of the integration domain tends to zero (seeDefinition 2.2). Our main result is as follows (see Theorem 2.5).

Assume that Ω is a finite rectangle in C with sides parallel to the real and imaginaryaxes. Let F (λ, μ) be an operator-valued function ranging in B(K,H), defined for allλ + iμ ∈ Ω, and satisfying the Lipschitz condition on Ω. The last property means thatthere exists a number γ1 > 0 such that

(1.4) ‖F (λ, μ)− F (λ′, μ′)‖ ≤ γ1(|λ− λ′|+ |μ− μ′|)

for any λ + iμ ∈ Ω and λ′ + iμ′ ∈ Ω. If, for some γ2 > 0, the function F satisfies theadditional condition

(1.5)∥∥F (λ, μ)− F (λ′, μ)− F (λ, μ′) + F (λ′, μ′)

∥∥ ≤ γ2 |λ− λ′| |μ− μ′|

for any λ+ iμ ∈ Ω and λ′ + iμ′ ∈ Ω, then the operator Stieltjes integral

(1.6)

∫Ω

F (Re z, Im z) dE(z)

exists in the sense of the uniform operator topology. If a B(H,K)-valued function Gdefined on Ω has properties (1.4) and (1.5), then a similar claim (see Corollary 2.7) holdsfor the operator Stieltjes integral

(1.7)

∫Ω

dE(z)G(Re z, Im z).

This result permits extending the notion of operator Stieltjes integral with respect to aspectral measure of a normal operator C to the case in which Ω = spec(C). Here one


OPERATOR STIELTJES INTEGRALS AND SOLUTIONS OF SOME EQUATIONS 47

should take into account the fact that only the values F (λ, μ) and G(λ, μ) correspondingto λ+ iμ ∈ spec(C) give a nontrivial contribution to (1.6) and (1.7).

Further, by using the operator Stieltjes integral, we find an integral representation ofthe solution X of the operator Sylvester equation

(1.8) XA− CX = D,

where A is an arbitrary densely defined closed operator on a Hilbert space H, C is anormal operator on a Hilbert space K, and D is a bounded operator from H to K. Inparticular, under the condition

(1.9) dist(spec(A), spec(C)

)> 0,

we prove that an operator X ∈ B(H,K) is a unique strong solution of (1.8) if and only ifit can be represented as the operator Stieltjes integral

(1.10) X =

∫spec(C)

dEC(z)D(A− z)−1

existing in the sense of the strong operator topology (see Theorem 4.5). Earlier, arepresentation of this kind for the solution of the Sylvester equation has only been provedfor the case in which C is a self-adjoint operator (see [4, Theorem 2.14]).

We also use the results obtained to study the operator Riccati equation

(1.11) XA− CX +XBX = D

under the assumption that the operators A, C, andD satisfy the same conditions as in thecase of the Sylvester equation (1.8) and B ∈ B(K,H). In particular, we use the fact thatif X ∈ B(H,K) is a strong solution of (1.11) such that dist

(spec(A+BX), spec(C)

)> 0,

then, in view of (1.10), the Riccati equation can be rewritten in the equivalent integralform

(1.12) X =

∫spec(C)

dEC(z)D(A+BX − z)−1.

Under condition (1.9) and some additional assumptions on the relative smallness of theoperatorsB andD, we prove that the integral equation (1.12) has a bounded solutionX ∈B(H,K). This solution is also a solution of the Riccati equation (1.11) (see Theorem 5.7).

Let us outline the structure of the paper.Section 2 introduces the notion of Stieltjes integral of an operator-valued function with

respect to the spectral measure of a normal operator. Here we also prove the main resulton sufficient conditions for the existence of such an integral (Theorem 2.5) and estimateits norm (Lemma 2.9).

Section 3 extends the notion [4, 23] of the norm of a bounded operator with respect toa spectral measure to the case in which this measure corresponds to a normal operator.

Section 4 discusses the operator Sylvester equation (1.8). In particular, we establishthat every weak solution of this equation is simultaneously a strong solution. Thisresult permits substantially strengthening the theorems in [4, Sec. 2] concerning therepresentation of the solution X of the Sylvester equation in the form of an operatorStieltjes integral. Moreover, the integral representation for X is extended to the case inwhich the coefficient C in (1.8) is a normal operator.

Finally, Section 5 states the above-mentioned result (Theorem 5.7) on the solvabilityconditions for the operator Riccati equation (1.11).

Let us introduce some notation to be used throughout the paper. For any Hilbertspace K, the identity operator on K is denoted by the same symbol I. If T is a closedoperator on K, then the symbol spec(T ) stands for the spectrum of T . We write

σ(T ) ={(λ, μ) ∈ R

2 |λ+ iμ ∈ spec(T )}



for the natural embedding of the set spec(T ) ∈ C in the real plane R2. The set

W(T ) = {z ∈ C | z = 〈Tx, x〉 for some x ∈ Dom(T ), ‖x‖ = 1}is called the numerical range of T .

2. Integral of an operator-valued function with respect to the spectral

measure of a normal operator

Let ABorel(C) be the algebra of Borel subsets of C and let {E(Ω)}Ω∈ABorel(C) be thespectral family of some (possibly unbounded) normal operator C. Recall that the op-erators E(Ω) are orthogonal projections on K with the following properties (e.g., see[7, Sec. 5.1]):

E(Ω′ ∩ Ω′′) = E(Ω′)E(Ω′′) for any Ω′,Ω′′ ∈ ABorel(C),(2.1)

E(Ω′ ∪ Ω′′) = E(Ω′) + E(Ω′′) if Ω′,Ω′′ ∈ ABorel(C) and Ω′ ∩ Ω′′ = ∅,(2.2)

and moreover,

(2.3) E(C) = E(spec(C)

)= I.

With the spectral family {E(Ω)}Ω∈ABorel(C), we associate a projection-valued measure

{E(Λ)}Λ∈ABorel(R2)

defined on the algebra ABorel(R2) of Borel subsets of R2 by setting

(2.4) E(Λ) = E({z = λ+ iμ | (λ, μ) ∈ Λ ⊂ R

2}).

Clearly, relations (2.1), (2.2), and (2.3) are equivalent to the relations

E(Λ′ ∩ Λ′′) = E(Λ′)E(Λ′′) for any Λ′,Λ′′ ∈ ABorel(R2),

E(Λ′ ∪ Λ′′) = E(Λ′) + E(Λ′′) if Λ′,Λ′′ ∈ ABorel(R2),Λ′ ∩ Λ′′ = ∅,

and

E(R2) = E(σ(C)

)= I,

respectively.In terms of the spectral measures E(·) and E(·), the spectral decomposition of the

normal operator C can be represented in the form

C =

∫C

z dE(z) =

∫spec(C)

z dE(z)

or

C =

∫R2

(λ+ iμ) dE(λ, μ) =

∫σ(C)

(λ+ iμ) dE(λ, μ).

We also need the projection-valued function E(λ, μ) defined on R2 by the formula

E(λ, μ) = E({(x, y) ∈ R

2 |x < λ, y < μ}).

In what follows, this function is referred to as the spectral function of the normal operatorC. Note that, in contrast to the case of self-adjoint operators, the spectral function of anormal operator is a function of two real variables rather than one.

It follows from the additivity of the spectral measure that

E(λ′, μ′) = E(λ, μ) + E((−∞, λ)× [μ, μ′)

)+ E([λ, λ′)× (−∞, μ)

)+ E([λ, λ′)× [μ, μ′)

)for λ ≤ λ′ and μ ≤ μ′. This relation obviously implies that

(2.5) E(λ, μ) ≤ E(λ′, μ′) if λ ≤ λ′ and μ ≤ μ′.

Thus, E(λ, μ) is a nondecreasing function of each of the arguments λ and μ.



It is easily seen that if λ ≤ λ′ and μ ≤ μ′, then

(2.6) E(λ′, μ′)− E(λ′, μ)− E(λ, μ′) + E(λ, μ) = E([λ, λ′)× [μ, μ′)

),

whence it follows that

(2.7) E(λ′, μ)− E(λ, μ) ≤ E(λ′, μ′)− E(λ, μ′)

for any μ, μ′ ∈ R such that μ ≤ μ′ and any λ, λ′ ∈ R such that λ ≤ λ′.The monotonicity (2.5) in particular implies that

(2.8) if λ′ ≤ λ and μ′ ≤ μ, then ‖E(λ′, μ′)x‖ ≤ ‖E(λ, μ)x‖ for each x ∈ K.

Indeed,

‖E(λ, μ)x‖2 = 〈E(λ, μ)x, E(λ, μ)x〉 = 〈E(λ, μ)x, x〉and

‖E(λ′, μ′)x‖2 = 〈E(λ′, μ′)x, E(λ′, μ′)x〉 = 〈E(λ′, μ′)x, x〉.Thus, (2.5) automatically implies (2.8).

Since both sides of (2.7) are orthogonal projections, we also conclude that if μ ≤ μ′

and λ ≤ λ′, then

(2.9)∥∥(E(λ′, μ)− E(λ, μ)

)x∥∥ ≤ ‖

(E(λ′, μ′)− E(λ, μ′)

)x‖

for every x ∈ K.In this section, we repeatedly make the following assumption.

Assumption 2.1. Assume that H and K are Hilbert spaces and that Δ is a rectangle inR2 of the form Δ = [a, b) × [c, d), where −∞ < a < b < +∞ and −∞ < c < d < +∞.Next, assume that {E(Λ)}Λ∈ABorel(R2) is the spectral family corresponding to some normaloperator on K.

Definition 2.2. Under Assumption 2.1, we say that an operator-valued function

F : Δ → B(K,H)

is uniformly (respectively, strongly or weakly) right integrable with respect to the spectralmeasure dE(λ, μ) on the set Δ if there exists a limit(2.10)∫

Δ

F (λ, μ) dE(λ, μ) = limmaxm

j=1 |δ(m)j |+maxn

k=1 |ω(n)k |→0

m∑j=1

n∑k=1

F (ξj , ζk)E(δ(m)j × ω

(n)k ),

in the sense of uniform (respectively, strong or weak) operator topology. Here δ(m)j =

[λj−1, λj), j = 1, 2, . . . ,m, and ω(n)k = [μk−1, μk), k = 1, 2, . . . , n, where a = λ0 < λ1 <

· · · < λm = b and c = μ0 < μ1 < . . . < μn = b are some partitions of the intervals [a, b)

and [c, d), respectively; next, ξj ∈ δ(m)j and ζk ∈ δ

(n)k are arbitrarily chosen points of the

intervals δ(m)j and δ

(n)k , and |δ(j)k | = λj − λj−1 and |ω(n)

k | = μk − μk−1 are the lengths

of these intervals. The limit value (2.10) of the integral sums, if it exists at all, is calledthe right Stieltjes integral of the operator-valued function F with respect to the measuredE(λ, μ) on the set Δ.

Likewise, an operator-valued function

G : Δ → B(H,K)



is said to be uniformly (respectively, strongly or weakly) left integrable with respect tothe spectral measure dE(λ, μ) on Δ if there exists a limit(2.11)∫

Δ

dE(λ, μ)G(λ, μ) = limmaxm

j=1 |δ(m)j |+maxn

k=1 |ω(n)k |→0

m∑j=1

n∑k=1

E(δ(m)j × ω

(n)k )G(ξj, ζk)

in the sense of uniform (respectively, strong or weak) operator topology. The limit value(2.11), if it exists at all, is called the left Stieltjes integral of the operator-valued functionG with respect to the spectral measure dE(λ, μ) on the set Δ.

The following claim extends [23, Lemma 10.5] to the case of the spectral measurecorresponding to a normal operator.

Lemma 2.3. Under Assumption 2.1, an operator-valued function F (λ, μ),

F : Δ → B(K,H),is left integrable with respect to the spectral measure dE(λ, μ) on Δ in the sense of weak(respectively, uniform) operator topology if and only if the function [F (λ, μ)]∗ is rightintegrable with respect to dE(λ, μ) on Δ in the same sense. Moreover, if this is the case,then

(2.12)

[∫Δ

F (λ, μ) dE(λ, μ)

]∗=

∫Δ

dE(λ, μ) [F (λ, μ)]∗.

Proof. By analogy with Lemma 10.5 in [23], this lemma can be proved by using thecontinuity of the involution T → T ∗ in the weak and uniform operator topologies onB(K,H). It suffices to apply this property to the integral sums in (2.10) and (2.11). �

Remark 2.4. Since the involution T → T ∗ is not continuous in the strong operatortopology (e.g., see [7, Sec. 2.5]), we see that, in general, the existence of one of theintegrals in (2.12) in the sense of this topology implies the existence of the other only inthe sense of the weak topology.

The following theorem describes sufficient conditions for the integrability of an operator-valued function F (λ, μ) with respect to the spectral measure of a normal operator.

Theorem 2.5. Let Assumption 2.1 hold. Assume also that, for some γ1, γ2 > 0, theoperator-valued function F : Δ → B(K,H) satisfies the inequalities

‖F (λ, μ)− F (λ′, μ′)‖ ≤ γ1 (|λ− λ′|+ |μ− μ′|),(2.13) ∥∥F (λ, μ)− F (λ′, μ)− F (λ, μ′) + F (λ′, μ′)∥∥ ≤ γ2 |λ− λ′| |μ− μ′|(2.14)

for any λ, λ′ ∈ [a, b) and μ, μ′ ∈ [c, d).Then the function F is right integrable with respect to the spectral measure dE(λ, μ)

on Δ in the sense of the uniform operator topology.

Proof. Let {δ(m)j }mj=1 and {ω(n)

k }nk=1 be partitions of the intervals [a, b) and [c, d), re-

spectively, and let Δ(m,n)jk = δ

(m)k × ω

(n)k , j = 1, 2, . . . ,m, k = 1, 2, . . . , n. Next,

let {(ξj , ζk) ∈ Δ(m,n)jk , j = 1, 2, . . . ,m, k = 1, 2, . . . , n} and {(ξ′j , ζ ′k) ∈ Δ

(m,n)jk , j =

1, 2, . . . ,m, k = 1, 2, . . . , n} be two sets of points arbitrarily chosen in the respective

rectangles Δ(m,n)jk . First, let us prove that the limit (2.10) (if it exists) is independent of

the choice of these points. Set

(2.15) Jm,n =

m∑j=1

n∑k=1

F (ξj , ζk)E(Δ(m,n)jk )



and

J ′m,n =

m∑j=1

n∑k=1

F (ξ′j , ζ′k)E(Δ

(m,n)jk ).

Note that, by (2.6),

(2.16) E(Δ(m,n)jk ) = E(λj , μk)− E(λj−1, μk)− E(λj , μk−1) + E(λj−1, μk−1)

and hence

Jm,n − J ′m,n =

m∑j=1

n∑k=1

[F (ξj , ζk)− F (ξ′j , ζ′k)]

×[E(λj , μk)− E(λj−1, μk)− E(λj , μk−1) + E(λj−1, μk−1)

].

Let us represent the difference Jm,n − J ′m,n as a sum of two easier-to-estimate terms,

(2.17) Jm,n − J ′m,n = L1 + L2,

where

L1 =m∑j=1

n∑k=1

[F (ξj , ζk)− F (ξ′j , ζk)]


]and

(2.18) L2 =m∑j=1

n∑k=1

[F (ξ′j , ζk)− F (ξ′j , ζ′k)]


].

A straightforward verification shows that

(2.19) L1 = S1 + S2 + S3,

where

S1 =

m∑j=1

[F (ξ′j , ζ1)− F (ξj , ζ1)

][E(λj , μ0)− E(λj−1, μ0)

],

S2 =

m∑j=1

[F (ξj , ζn)− F (ξ′j , ζn)

][E(λj , μn)− E(λj−1, μn)

],

and

S3 =m∑j=1

n−1∑k=1

[F (ξj , ζk)− F (ξ′j , ζk)− F (ξj , ζk+1) + F (ξ′j , ζk+1)

]×[E(λj , μk)− E(λj−1, μk)

].

Since the operator-valued function F satisfies the Lipschitz condition (2.13) for everyx ∈ K, we have

‖S1x‖ ≤ γ1

m∑j=1

|δ(m)j |

∥∥(E(λj , μ0)− E(λj−1, μ0))x∥∥,



whence it follows that

‖S1x‖ ≤ γ1

(m∑j=1

|δ(m)j |2

)1/2( m∑j=1

⟨(E(λj , μ0)− E(λj−1, μ0)

)x, x⟩)1/2

≤ γ1

(m

maxj=1

|δ(m)j |

)1/2√b− a

(⟨(E(b, μ0)− E(a, μ0)

)x, x⟩)1/2

(2.20)

≤ γ1

(m

maxj=1

|δ(m)j |

)1/2√b− a ‖E(b, c)− E(a, c)‖ ‖x‖,

because⟨(E(b, μ0)− E(a, μ0)

)x, x⟩=⟨(E(b, μ0)− E(a, μ0)

)2x, x⟩

=⟨(E(b, μ0)− E(a, μ0)

)x,(E(b, μ0)− E(a, μ0)

)x⟩

(2.21)

=∥∥(E(b, μ0)− E(a, μ0)

)x∥∥2

and μ0 = c. In a completely similar way, we show that

(2.22) ‖S2x‖ ≤ γ1

(m

maxj=1

|δ(m)j |

)1/2√b− a ‖E(b, d)− E(a, d)‖ ‖x‖

for every x ∈ K. The norm of S3x, x ∈ K, can be estimated as follows:

‖S3x‖ ≤ γ2

m∑j=1

n−1∑k=1

|δ(m)j | |ω(n)

k |∥∥(E(λj , μk)− E(λj−1, μk)

)x∥∥

= γ2

n−1∑k=1

|ω(n)k |

m∑j=1

|δ(m)j |

∥∥(E(λj , μk)− E(λj−1, μk))x∥∥

≤ γ2

n−1∑k=1

|ω(n)k |

(m∑j=1

|δ(m)j |2

)1/2( m∑j=1

⟨(E(λj , μk)− E(λj−1, μk)

)x, x⟩)1/2

≤ γ2 (d− c)(

mmaxj=1

|δ(m)j |

)1/2√b− a

n−1maxk=1

(⟨(E(b, μk)− E(a, μk)

)x, x⟩)1/2

≤ γ2√b− a (d− c)

(m

maxj=1

|δ(m)j |

)1/2 n−1maxk=1

‖E(b, μk)− E(a, μk)‖ ‖x‖.

Here we use property (2.14) at the first step and the result (2.21) (with μ0 replaced byμk) at the last step. Obviously, inequality (2.7) implies that

E(b, μk)− E(a, μk) ≤ E(b, μn)− E(a, μn) = E(b, d)− E(a, d)

for all k = 1, 2, . . . , n− 1. Thus, we arrive at the following definitive estimate for S3:

(2.23) ‖S3x‖ ≤ γ2√b− a (d− c)

(m

maxj=1

|δ(m)j |

)1/2‖E(b, d)− E(a, d)‖ ‖x‖.

By combining the estimates (2.19), (2.20), (2.22), and (2.23), we find that

(2.24) ‖L1x‖ ≤ M1

(m

maxj=1

|δ(m)j |

)1/2 √b− a ‖x‖

for every x ∈ K, where

M1 = γ1‖E(b, c)− E(a, c)‖+(γ1 + γ2(d− c)

)‖E(b, d)− E(a, d)‖.



The term L2 given by (2.18) can be estimated by the same scheme. The resultinginequality has the following form, similar to (2.24):

(2.25) ‖L2x‖ ≤ M2

(n

maxk=1

|ω(n)k |)1/2 √

d− c ‖x‖,

where

M2 = γ1‖E(a, d)− E(a, c)‖+(γ1 + γ2(b− a)

)‖E(b, d)− E(b, c)‖.

Relations (2.17), (2.24), and (2.25) taken together prove that if the sum on the right-hand side in (2.10) converges strongly (respectively, weakly or in the uniform operator

topology) for some choice of the points {ξj ∈ δ(m)j }mj=1 and {ζk ∈ ω

(n)k }nk=1, then it

converges strongly (respectively, weakly or in the uniform operator topology) to the samelimit value for any other choice of these points. For these points we can, in particular,take the left ends

(2.26) ξj = λj−1, j = 1, 2, . . . ,m, and ζk = μj−1, k = 1, 2, . . . , n,

of the corresponding partition intervals. In what follows, we show that, for this (andhence for any other) choice of the points ξj and ζk, the double sequence of operators

Jm,n given by (2.15) indeed has a limit as m → ∞, n → ∞, maxmj=1 |δ(m)j | → 0, and

maxnk=1 |ω(n)k | → 0.

Consider two distinct partitions of [a, b) into m and m > m subintervals, respectively.Consider also two distinct partitions of [c, d) into n and n > n subintervals, respectively.Without loss of generality, we additionally assume that all points of the m-partitionof [a, b) are simultaneously points of the m-partition of [a, b) and that all points ofthe n-partition of [c, d) are simultaneously points of the n-partition of [c, d). Let λj,s,s = 0, 1, . . . , pj , be the points of the m-partition of [a, b) lying in the subinterval [λj−1, λj ],and let μk,t, t = 0, 1, . . . , qk, be the points of the n-partition of [c, d) that lie in thesubinterval [μk−1, μk]. Thus, we have

λj−1 = λj,0 < λj,1 < · · · < λj,pj= λj , j = 1, 2, . . . ,m,(2.27)

μk−1 = μk,0 < μk,1 < · · · < μk,qk = μk, k = 1, 2, . . . , n.(2.28)

Assuming that the points ξj , ζk, ξj,s, and ζk,t are chosen according to the rule (2.26)as the left points of subintervals of the corresponding partitions, we obtain

(2.29) Jm,n =

m∑j=1

n∑k=1

F (λj−1, μk−1)E(Δ(m,n)jk )

and

(2.30) Jm,n =

m∑j=1

n∑k=1

pj∑s=1

qk∑t=1

F (λj,s−1, μk,t−1)E(Δ(m,n)j,s;k,t),

where

Δ(m,n)j,s;k,t = [λj,s−1, λj,s)× [μk,t−1;k,t).

By using (2.16), we rewrite the integral sums (2.29) and (2.30) in the form

(2.31) Jm,n =

m∑j=1

n∑k=1

F (λj−1, μk−1)


]



and

(2.32) Jm,n =

m∑j=1

n∑k=1

pj∑s=1

qk∑t=1

F (λj,s−1, μk,t−1)

×[E(λj,s, μk,t)− E(λj,s−1, μk,t)− E(λj,s, μk,t−1) + E(λj,s−1, μk,t−1)

],

respectively. In view of (2.27) and (2.28), a straightforward verification shows that

(2.33) Jm,n − Jmn =

m∑j=1

n∑k=1

(T(1)jk + T

(2)jk + T

(3)jk ),

where

T(1)jk =

pj−1∑s=1

[F (λj,s, μk,0)− F (λj,s−1, μk,0)

]×[E(λj,pj

, μk,qk)− E(λj,s, μk,qk)− E(λj,pj, μk,0) + E(λj,s, μk,0)

]=

pj−1∑s=1

[F (λj,s, μk−1)− F (λj,s−1, μk−1)

]×[E(λj , μk)− E(λj,s, μk)− E(λj , μk−1) + E(λj,s, μk−1)

],

T(2)jk =

qk−1∑t=1

[F (λj,0, μk,t)− F (λj,0, μk,t−1)

]×[E(λj,pj

, μk,qk)− E(λj,0, μk,qk)− E(λj,pj, μk,t) + E(λj,0, μk,t)

]=

qk−1∑t=1

[F (λj−1, μk,t)− F (λj−1, μk,t−1)

]×[E(λj , μk)− E(λj−1, μk)− E(λj , μk,t) + E(λj−1, μk,t)

],

and

(2.34) T(3)jk =

pj−1∑s=1

qk−1∑t=1

[F (λj,s, μk,t)− F (λj,s−1, μk,t)− F (λj,s, μk,t−1)

+ F (λj,s−1, μk,t−1)][E(λj , μk)− E(λj,s, μk)− E(λj , μk,t) + E(λj,s, μk,t)

].

Note also that

n∑k=1

T(1)jk =

pj−1∑s=1

{[F (λj,s, μn−1)− F (λj,s−1, μn−1)

][E(λj , μn)− E(λj,s, μn)

](2.35)

−[F (λj,s, μ0)− F (λj,s−1, μ0)

][E(λj , μ0)− E(λj,s, μ0)

]+

n−1∑k=1

[F (λj,s, μk−1)− F (λj,s−1, μk−1)− F (λj,s, μk) + F (λj,s−1, μk)

]×[E(λj , μk)− E(λj,s, μk)

]}



and

m∑j=1

T(2)jk =

qk−1∑t=1

{[F (λm−1, μk,t)− F (λm−1, μk,t−1)

][E(λm, μk)− E(λm, μk,t)

]−[F (λ0, μk,t)− F (λ0, μk,t−1)

][E(λ0, μk)− E(λ0, μk,t)

]+

m−1∑j=1

[F (λj−1, μk,t)− F (λj−1, μk,t−1)− F (λj , μk,t) + F (λj , μk,t−1)

]×[E(λj , μk)− E(λj , μk,t)

]}.

In view of (2.13) and (2.14), relation (2.35) implies that, for every x ∈ K,∥∥∥∥∥n∑

k=1

T(1)jk x

∥∥∥∥∥ ≤pj−1∑s=1

{γ1 |λj,s − λj,s−1|

∥∥(E(λj , μn)− E(λj,s, μn))x∥∥

+ γ1 |λj,s − λj,s−1|∥∥(E(λj , μ0)− E(λj,s, μ0)

)x∥∥

+

n−1∑k=1

γ2 |λj,s − λj,s−1| |ω(n)k |

∥∥(E(λj , μk)− E(λj,s, μk))x∥∥}

and hence ∥∥∥∥∥n∑

k=1

T(1)jk x

∥∥∥∥∥ ≤ γ1 |δ(m)j |

pj−1maxs=1

∥∥(E(λj , d)− E(λj,s, d))x∥∥(2.36)

+ γ1 |δ(m)j |

pj−1maxs=1

∥∥(E(λj , c)− E(λj,s, c))x∥∥

+ γ2 |δ(m)j |(d− c)| n−1

maxk=1

pj−1maxs=1

∥∥(E(λj , μk)− E(λj,s, μk))x∥∥,

because μ0 = c, μn = d, and moreover,

pj−1∑s=1

|λj,s − λj,s−1| = λj,pj−1− λj−1 < λj − λj−1 = |δ(m)

j |,

while∑n−1

k=1 |ω(n)k | < d− c. Clearly,∥∥(E(λj , μ)− E(λj,s, μ)

)x∥∥ ≤

∥∥(E(λj , μ)− E(λj−1, μ))x∥∥(2.37)

for all μ ∈ R, because∥∥(E(λj , μ)− E(λj,s, μ))x∥∥2 =

⟨E([λj,s, λj)× (−∞, μ)

)x, x⟩

≤⟨E([λj−1, λj)× (−∞, μ)

)x, x⟩

=∥∥(E(λj , μ)− E(λj−1, μ)

)x∥∥2.

By using inequalities (2.37) and (2.9), we find from (2.36) that∥∥∥∥∥n∑

k=1

T(1)jk x

∥∥∥∥∥ ≤ [2γ1 + γ2(d− c)] |δ(m)j |

∥∥(E(λj , d)− E(λj−1, d))x∥∥.



Thus, ∥∥∥∥∥m∑j=1

n∑k=1

T(1)jk x

∥∥∥∥∥ ≤ [2γ1 + γ2(d− c)]m∑j=1

|δ(m)j |

∥∥(E(λj , d)− E(λj−1, d))x∥∥(2.38)

≤ [2γ1 + γ2(d− c)]

(m∑j=1

|δ(m)j |2

)1/2( m∑j=1

∥∥(E(λj , d)− E(λj−1, d))x∥∥2)1/2

≤ [2γ1 + γ2(d− c)](

mmaxj=1

|δ(m)j |

)1/2( m∑j=1

|δ(m)j |

)1/2

×(

m∑j=1

⟨E(λj , d)− E(λj−1, d)x, x

⟩)1/2

≤ [2γ1 + γ2(d− c)](

mmaxj=1

|δ(m)j |

)1/2√b− a

(⟨E(b, d)− E(a, d)x, x

⟩)1/2≤ [2γ1 + γ2(d− c)]

( mmaxj=1

|δ(m)j |

)1/2√b− a ‖E(b, d)− E(a, d)‖ ‖x‖.

In exactly the same way, we prove that∥∥∥∥∥m∑j=1

T(2)jk x

∥∥∥∥∥ ≤ [2γ1 + γ2(b− a)] |ω(n)k |

∥∥(E(b, μk)− E(b, μk−1))x∥∥

and hence

(2.39)

∥∥∥∥∥m∑j=1

n∑k=1

T(2)jk x

∥∥∥∥∥ ≤ [2γ1+γ2(b−a)](

nmaxk=1

|ω(n)k |)1/2√

d− c ‖E(b, d)− E(b, c)‖ ‖x‖.

It remains to estimate the contribution to the difference (2.33) from the operators T(3)jk

defined in (2.34). We use identity (2.6) and inequality (2.14) and readily see that (2.34)implies that∥∥T (3)

jk x∥∥ ≤ γ2

pj−1∑s=1

qk−1∑t=1

|λj,s − λj,s−1| |μk,t − μk,t−1|∥∥E([λj,s, λj)× [μk,t, μk)

)x∥∥(2.40)

≤ γ2

pj−1∑s=1

qk−1∑t=1

|λj,s − λj,s−1| |μk,t − μk,t−1| ‖E(δ(m)j × ω

(n)k )x‖

≤ γ2 |λj,pj−1 − λj−1| |μk,qk−1 − μk−1| ‖E(δ(m)j × ω

(n)k )x‖

≤ γ2 |δ(m)j | |ω(n)

k | ‖E(δ(m)j × ω

(n)k )x‖ = γ2 |δ(m)

j | |ω(n)k | ‖E(Δ(m,n)

jk )x‖.Consequently,

∥∥∥∥∥m∑j=1

n∑k=1

T(3)jk x

∥∥∥∥∥ ≤m∑j=1

n∑k=1

‖T (3)jk x‖

(2.41)

≤ γ2

(m∑j=1

n∑k=1

|δ(m)j |2 |ω(n)

k |2)1/2( m∑

j=1

n∑k=1

‖E(Δ(m,n)jk )x‖2

)1/2

= γ2

(m∑j=1

|δ(m)j |2

)1/2( n∑k=1

|ω(n)k |2

)1/2( m∑j=1

n∑k=1

〈E(Δ(m,n)jk )x, x〉

)1/2



≤ γ2

(m

maxj=1

|δ(m)j | n

maxk=1

|ω(n)k |)1/2√

(b− a)(d− c) 〈E(Δ)x, x〉1/2

≤ γ2

(m

maxj=1

|δ(m)j | n

maxk=1

|ω(n)k |)1/2√

(b− a)(d− c) ‖E(Δ)‖ ‖x‖.

In view of (2.33), inequalities (2.38), (2.39), and (2.41) prove that the integral sum (2.29)indeed converges as m → ∞, n → ∞, maxmj=1 |δmj | → 0, and maxnk=1 |ωn

k | → 0 to someoperator in B(K,H). Moreover, the convergence holds in the uniform operator topology.

The proof is complete. �

Remark 2.6. The question as to whether the sufficient conditions (2.13) and (2.14) forthe integrability of an operator-valued function with respect to the spectral measure of anormal operator in the sense of the uniform operator topology are optimal remains open.We point out that a similar problem has not been solved even in the simpler case wherethe spectral family {E(·)} corresponds to a self-adjoint operator. Namely, it is knownthat if an operator-valued function defined on a finite interval in R is Lipschitz, then itis integrable on that interval in the sense of the uniform operator topology with respectto the spectral measure of any self-adjoint operator (see [1, Lemma 7.2 and Remark7.3]). However, as far as we know, it has not been established yet whether the Lipschitzcondition is necessary indeed.

Corollary 2.7. Assume that an operator-valued function G : Δ → B(H,K) satisfies con-ditions (2.13) and (2.14) for some γ1, γ2 > 0. Then G is left integrable on Δ with respectto the spectral measure dE(λ, μ), and the integral (2.11) exists in the sense of the uniformoperator topology.

Proof. In view of Lemma 2.3, this is a straightforward consequence of Theorem 2.5. �

Remark 2.8. If the integral (2.11) exists (as an operator that is an element of B(H,K)),then its range is necessarily contained in the subspace KΔ = Ran

(E(Δ)

).

A norm estimate for the integral of an operator-valued function with respect to thespectral measure of a normal operator is given in the following lemma.

Lemma 2.9. Under the assumptions of Theorem 2.5, the norm of the integral (2.10)viewed as an operator in B(K,H) satisfies the estimate

(2.42)

∫Δ

F (λ, μ) dE(λ, μ)

≤ 4 sup(λ,μ)∈Δ

‖F (λ, μ)‖+ 2γ1 (b− a+ d− c) + γ2(b− a)(d− c)].

Proof. Let points a = λ0 < λ1 < · · · < λm = b and c = μ0 < μ1 < · · · < μn = d

specify partitions of the intervals [a, b) and [c, d), respectively. Let δ(m)j = [λj−1, λj),

j = 1, 2, . . . ,m, and ω(n)k = [μk−1, μk), k = 1, 2, . . . , n, be the elementary intervals of

these partitions. Set Δ(m,n)jk = δ

(m)k × ω

(n)k and note that the integral sum

(2.43) Jm,n =

m∑j=1

n∑k=1

F (λj−1, μk−1)E(Δ(m,n)jk )

can be rewritten in the form

Jm,n =

m∑j=1

n∑k=1

F (λj−1, μk−1)[E(λj , μk)− E(λj−1, μk)− E(λj , μk−1) + E(λj−1, μk−1)

].



By rearranging the terms and by taking into account the relations λ0 = a, λm = b,μ0 = c, and μn = d, we can represent this sum as

Jm,n = S1 + S2 + S3 + S4,(2.44)

where

S1 = F (λm−1, μn−1)E(b, d)− F (a, μn−1)E(a, d)− F (λm−1, c)E(b, c) + F (a, c)E(a, c),

S2 =

m−1∑j=1

{[F (λj−1, μn−1)− F (λj , μn−1)

]E(λj , d)−

[F (λj−1, c)− F (λj , c)

]E(λj , c)

},

S3 =n−1∑k=1

{[F (λm−1, μk−1)−F (λm−1, μk)

]E(b, μk)−

[F (a, μk−1)−F (a, μk)

]E(a, μk)

},

and

S4 =m−1∑j=1

n−1∑k=1

[F (λj , μk)− F (λj−1, μk)− F (λj , μk−1) + F (λj−1, μk−1)

]E(λj , μk).

Since

(2.45) ‖E(λ, μ)‖ ≤ 1 for any λ, μ ∈ R,

we obviously have

S1 ≤ 4 sup(λ,μ)∈Δ

‖F (λ, μ)‖.(2.46)

By assumption, the operator-valued function F satisfies the estimates (2.13) and (2.14),which, in view of (2.45), imply that

S2 ≤ 2γ1

m−1∑j=1

|λj−1 − λj | < 2γ1(b− a),(2.47)

S3 ≤ 2γ1

n−1∑k=1

|μk−1 − μk| < 2γ1(d− c),(2.48)

and

S4 ≤ γ2

m−1∑j=1

n−1∑k=1

|λj−1 − λj | |μk−1 − μk| < γ2(b− a)(d− c).(2.49)

Relations (2.44) and (2.46)–(2.49) taken together show that, after passing to the limit as

maxmj=1 |δ(m)j |+maxnk=1 |ω

(n)k | → 0 in (2.43), we arrive at inequality (2.42). The proof of

the lemma is complete. �In what follows, we assume that Ω = {z ∈ C | a ≤ Re z < b, c ≤ Im z < d} is a

rectangle on the complex plane C; here a, b, c, and d are finite. Let F : Ω → B(K,H)and G : Ω → B(H,K) be operator-valued functions of the complex variable z such thatthe corresponding functions F (λ+ iμ) and G(λ+ iμ) of the real variables λ ∈ [a, b) andμ ∈ [c, d) are right and left integrable, respectively, with respect to the spectral measuredE(λ, μ) of some normal operator C on the rectangle Δ = [a, b)× [c, d) in the real planeR

2. In this case, we set ∫Ω

F (z) dE(z) =

∫Δ

F (λ+ iμ) dE(λ, μ),(2.50) ∫Ω

dE(z)G(z) =

∫Δ

dE(λ, μ)G(λ+ iμ)(2.51)



by definition, where dE(z) stands for the spectral measure dE(λ, μ) transferred to thecomplex plane according to the rule (2.4).

Let F : Ω → B(K,H) and G : Ω → B(H,K) be such that the corresponding operator-valued functions F (λ+iμ) and G(λ+iμ) of the real variables (λ, μ) ∈ Δ satisfy conditions(2.13) and (2.14) on Δ. Then the integrals (2.50) and (2.51) exist by Theorem 2.5 andCorollary 2.7, respectively. Clearly, in view of (2.10) and (2.11), only the values of Fand G at the points lying in the support of dE, or, which is the same, belonging to thespectrum of C, give a nontrivial contribution to these integrals. Hence the correspondingintegrals of F and G over the spectral measure dE on the part of the spectrum of C lyingin Ω are defined by the formulas∫

spec(C)∩Ω

F (z)dE(z) =

∫Ω

F (z)dE(z),(2.52) ∫spec(C)∩Ω

dE(z)G(z) =

∫Ω

dE(z)G(z).(2.53)

In particular, if a function F : spec(C) → B(K,H) (respectively, G : spec(C) →B(H,K)) defined (only) on the spectrum of C admits a continuation F (respectively, G)

into the entire Ω such that conditions (2.13) and (2.14) hold for F (λ+ iμ) (respectively,

for G(λ+ iμ)) for all (λ, μ) ∈ Δ, then we set∫spec(C)∩Ω

F (z)dE(z) =

∫Ω

F (z)dE(z),(2.54) ∫spec(C)∩Ω

dE(z)G(z) =

∫Ω

dE(z)G(z).(2.55)

Clearly, the values (2.54) and (2.55) are independent of the choice of the continuations

F (z) and G(z) (at least if F (λ+iμ) and G(λ+iμ) treated as functions of the real variablesλ and μ have properties (2.13) and (2.14)).

Finally, the improper weak, strong, or uniform integrals

(2.56)

∫ b

a

∫ d

c

F (λ, μ) dE(λ, μ) and

∫ b

a

∫ d

c

dE(λ, μ)G(λ, μ)

with infinite lower and/or upper limits (a = −∞ and/or b = +∞ and c = −∞ and/ord = +∞) are understood as the limit values (if these exist) of the integrals with finiteintegration limits in the corresponding topologies. For example,∫ ∞

−∞

∫ ∞

−∞dE(λ, μ)G(λ, μ) = lim

a↓−∞, b↑∞c↓−∞, d↑∞

∫ b

a

∫ d

c

dE(λ, μ)G(λ, μ).

If dE(z) (or, which is the same, dE(λ, μ)) is the spectral measure corresponding to anunbounded normal operator C, then the corresponding integrals over the spectrum of Care understood as the improper integrals∫

spec(C)

F (z) dE(z) =

∫ +∞

−∞

∫ +∞

−∞F (λ+ iμ) dE(λ, μ)

and ∫spec(C)

dE(z)G(z) =

∫ +∞

−∞

∫ +∞

−∞dE(λ, μ) G(λ+ iμ).

Here it is assumed that the continuations F (λ+iμ) and G(λ+iμ) of the operator-valuedfunctions F : spec(C) → B(K,H) and G : spec(C) → B(H,K) should satisfy conditions(2.13) and (2.14) on every finite rectangle Δ = [a, b)× [c, d).



In conclusion of the section, we present the following natural result.

Lemma 2.10. Let H and K be Hilbert spaces. Assume that C is a normal (possiblyunbounded) operator on K, and let dE(z) be the corresponding spectral measure on C.Let Ω = {z ∈ C | a ≤ Re z < b, c ≤ Im z < d} be a rectangle in C with finite a, b, c,and d. Furthermore, let G(z) be a B(H,K)-valued function defined and holomorphic inan open disk in C entirely containing the closure Ω of Ω. Then

(2.57) Ran

(∫Ω

dE(z)G(z)

)⊂ Dom(C)

and

(2.58)

∫Ω

dE(z)(zG(z)

)= C

∫Ω

dE(z)G(z).

Proof. The inclusion (2.57) follows from definition (2.51) of the right integral of anoperator-valued function with respect to the spectral measure dE(z) and from Re-mark 2.8.

By assumption, the functions G(z) and zG(z) are holomorphic on Ω. Hence, beingrewritten in the variables λ = Re z ∈ [a, b) and μ = Im z ∈ [c, d), both functions auto-matically satisfy conditions (2.13) and (2.14). But then, by Theorem 2.5, the integrals∫

Ω

dE(z)G(z) and

∫Ω

dE(z)(z G(z)

)exist in the sense of the uniform operator topology.

Assume that the open disk mentioned in the lemma has center z0 and radius r. Wedenote this disk by Cr(z0) and write out the Taylor formula for the function G(z) withrespect to the point z0,

(2.59) G(z) =

n∑k=0

Gk(z − z0)k +Rn(z), z ∈ Cr(z0),

where Gk = G(k)(z0)k! and Rn(z) is the remainder term. Since G(z) is holomorphic on

Cr(z0) and Ω is a compact set contained in Cr(z0), it follows thatRn(z) and its derivativesconverge to zero as n → ∞, the convergence being uniform with respect to z ∈ Ω. Inparticular, it follows that

supλ+iμ∈Ω

|Sn(λ, μ)| −−−−→n→∞

0,(2.60)

supλ+iμ∈Ω

max

{∣∣∣∣∂Sn(λ, μ)

∂λ

∣∣∣∣, ∣∣∣∣∂Sn(λ, μ)

∂μ

∣∣∣∣} −−−−→n→∞

0,(2.61)

and

(2.62) supλ+iμ∈Ω

max

{∣∣∣∣∂2Sn(λ, μ)

∂λ2

∣∣∣∣, ∣∣∣∣∂2Sn(λ, μ)

∂μ2

∣∣∣∣, ∣∣∣∣∂2Sn(λ, μ)

∂λ∂μ

∣∣∣∣} −−−−→n→∞

0,

whereSn(λ, μ) = zRn(z) = (λ+ iμ)Rn(λ+ iμ), λ ∈ [a, b), μ ∈ [c, d).

It follows from (2.61) and (2.62) that there exists a sequence γn > 0, n = 0, 1, 2, . . . , suchthat

(2.63) γn → 0, as n → ∞and

(2.64)‖Sn(λ, μ)− Sn(λ

′, μ′)‖ ≤ γn (|λ− λ′|+ |μ− μ′|),‖Sn(λ, μ)− Sn(λ

′, μ)− Sn(λ, μ′) + Sn(λ

′, μ′)‖ ≤ γn |λ− λ′| |μ− μ′|



for any λ, λ′ ∈ [a, b) and μ, μ′ ∈ [c, d). By Corollary 2.7 and Lemma 2.9, it follows from(2.60), (2.63), and (2.64) that

(2.65)

∥∥∥∥∥∫Ω

dE(z)zRn(z)

∥∥∥∥∥ −−−−→n→∞

0.

A similar argument shows that∥∥∥∥∥∫Ω

dE(z)Rn(z)

∥∥∥∥∥ −−−−→n→∞

0

and hence

(2.66)

∥∥∥∥∥C∫Ω

dE(z)Rn(z)

∥∥∥∥∥ =

∥∥∥∥∥(CE(Ω)) ∫

Ω

dE(z)Rn(z)

∥∥∥∥∥ −−−−→n→∞

0,

because the product CE(Ω) is a bounded operator.By (2.59), we have∫

Ω

dE(z)(z G(z)

)=

n∑k=0

(∫Ω

z(z − z0)kdE(z)

)Gk +

∫Ω

dE(z)zRn(z).

Since RanE(Ω) ⊂ Dom(Cl) for any l = 1, 2, . . ., it follows that∫Ω

dE(z)(z G(z)

)=

n∑k=0

C(C − z0)kE(Ω)Gk +

∫Ω

dE(z)zRn(z)(2.67)

= Cn∑

k=0

(∫Ω

(z − z0)kdE(z)

)Gk +

∫Ω

dE(z)zRn(z).

On the other hand, it also follows from (2.59) that

C

∫Ω

dE(z)G(z) = C

n∑k=0

(∫Ω

(z − z0)kdE(z)

)Gk + C

∫Ω

dE(z)Rn(z).

By comparing the last equation with (2.67), we conclude that∫Ω

dE(z)(z G(z)

)− C

∫Ω

dE(z)G(z) =

∫Ω

dE(z)zRn(z)− C

∫Ω

dE(z)Rn(z).

To complete the proof, it remains to use the limit relations (2.65) and (2.66). �

3. Norm of an operator with respect to a spectral measure

The paper [23] introduced the notion of norm of a bounded operator with respectto the spectral measure of a self-adjoint operator; this notion proved quite useful whenstudying operator Sylvester and Riccati equations (see [4]). The aim of this section is togeneralize this notion to the case in which the spectral measure corresponds to a normaloperator.

Definition 3.1. Let Y ∈ B(H,K) be a bounded operator from a Hilbert space H toa Hilbert space K, and let {E(Ω)}Ω∈ABorel(C) be the spectral family of some (possiblyunbounded) normal operator on K. The number

(3.1) ‖Y ‖E =

(sup{Ωk}

∑k

‖Y ∗E(Ωk)Y ‖)1/2

,

where the supremum is taken over all finite (or countable) systems of pairwise disjointBorel subsets Ωk of the complex plane C, Ωk ∩Ωl = ∅ for k �= l, is called the norm of the



operator Y with respect to the spectral measure dE(z), or simply the E-norm of Y . ForZ ∈ B(K,H), the E-norm ‖Z‖E is defined by the formula ‖Z‖E = ‖Z∗‖E .

It is easily seen that if the E-norm ‖Y ‖E is finite, then

‖Y ‖ ≤ ‖Y ‖E .

If, moreover, Y is a Hilbert–Schmidt operator, then

(3.2) ‖Y ‖E ≤ ‖Y ‖2, Y ∈ B2(H,K),

where ‖ · ‖2 stands for the (Hilbert–Schmidt) norm on the ideal B2(H,K) of Hilbert–Schmidt operators from H to K.

The following claim generalizes Lemma 10.7 in [23].

Lemma 3.2. Let C be a normal operator on a Hilbert space K. Assume that an operator-valued function F : spec(C) → B(H) is bounded,

‖F‖∞ = supz∈spec(C)

‖F (z)‖ < ∞,

and, treated as a function of the variables λ = Re z and μ = Im z, admits a boundedcontinuation from σ(C) to the entire real plane R2 such that conditions (2.13) and (2.14)are satisfied on each finite rectangle in R2. If the E-norm ‖Y ‖E of an operator Y ∈B(H,K) with respect to the spectral measure dE(z) on C corresponding to C is finite,then the integrals

∫spec(C)

dE(z)Y F (z) and∫spec(C)

F (z)Y ∗ dE(z) exist in the sense of

uniform operator topology and satisfy the estimates∥∥∥∫spec(C)

dE(z)Y F (z)∥∥∥ ≤ ‖Y ‖E · ‖F‖∞,(3.3) ∥∥∥∫

spec(C)

F (z)Y ∗ dE(z)∥∥∥ ≤ ‖Y ‖E · ‖F‖∞.(3.4)

Proof. Let us carry out the proof using the integral in (3.3) as an example. We denotea continuation of the function F (λ+ iμ) from σ(C) to R2 by F (λ, μ), λ, μ ∈ R.

Let [a, b) ⊂ R and [c, d) ⊂ R be finite intervals (i.e., −∞ < a < b < ∞ and −∞ <

c < d < ∞) and let{δ(m)j

}mj=1

and{ω(n)k

}nk=1

be some partitions of [a, b) and [c, d),

respectively. If(δ(m)j × ω

(n)k

)∩ σ(C) �= ∅, then we take points ξj ∈ δ

(m)j and ζk ∈ ω

(n)k

such that ξj+iζk ∈ spec(C), or, equivalently, (ξj , ζk) ∈ σ(C). If(δ(m)j ×ω

(n)k

)∩σ(C) = ∅,

then we do not subject the choice of ξj ∈ δ(m)j and ζk ∈ ω

(n)k to any constraints. Since

E(δ(m)j × ω

(n)k )E(δ

(m)s × ω

(n)t ) = 0, whenever j �= s or k �= t, we see that∥∥∥∥∥

m∑j=1

n∑k=1

E(δ(m)j × ω

(n)k )Y F (ξj , ζk)x

∥∥∥∥∥2

=

⟨m∑j=1

n∑k=1

E(δ(m)j × ω

(n)k )Y F (ξj , ζk)x,

m∑s=1

n∑t=1

E(δ(m)s × ω

(n)t )Y F (ξs, ζt)x

⟩

=

⟨m∑j=1

n∑k=1

[F (ξj , ζk)]∗Y ∗E(δ

(m)j × ω

(n)k )Y F (ξj , ζk)x, x

⟩

=

⟨ ∑j,k : (ξj ,ζk)∈σ(C)

[F (ξj , ζk)]∗Y ∗E(δ

(m)j × ω

(n)k )Y F (ξj , ζk)x, x

⟩



for every x ∈ H. Consequently,∥∥∥∥∥m∑j=1

n∑k=1

E(δ(m)j × ω

(n)k )Y F (ξj , ζk)x

∥∥∥∥∥2

(3.5)

≤ sup(ξ,ζ)∈σ(C)

‖F (ξ, ζ)‖2∑

j,k : (ξj ,ζk)∈σ(C)∩Δ

‖Y ∗E(δ(m)j × ω

(n)k )Y ‖ ‖x‖2

≤ ‖F‖2∞‖Y ‖2E,Δ ‖x‖2,

where Δ = [a, b)× [c, d),

(3.6) ‖Y ‖E,Δ =

(sup{Δk}

∑k

‖Y ∗E(Δk)Y ‖)1/2

,

and the supremum in (3.6) is taken over all finite (or countable) systems of pairwisedisjoint Borel subsets Δk of the rectangle Δ. Obviously,

(3.7) ‖Y ‖E,Δ ≤ ‖Y ‖Eand

(3.8) lima↓−∞, b↑∞c↓−∞, d↑∞

‖Y ‖E,Δ = ‖Y ‖E .

By assumption, the function F (λ, μ) satisfies all conditions of Theorem 2.5 on everyfinite triangle Δ. Consequently, it is right integrable on each such Δ with respect to themeasure dE(λ, μ) by Corollary 2.7. It follows from (3.5) that∥∥∥∥∥

∫Δ

dE(λ, μ)Y F (λ, μ)

∥∥∥∥∥ ≤ ‖F‖∞‖Y ‖E,Δ.(3.9)

Then the existence of the limit (3.8) implies that∥∥∥∥∥∫[a′,b′)×[c′,d′)


∥∥∥∥∥→ 0

as a′, b′ → ∞ or a′, b′ → −∞ and/or c′, d′ → ∞ or c′, d′ → −∞.In view of (3.9), this implies the existence of a limit∫

R2

dE(λ, μ)Y F (λ, μ) = n-lima↓−∞, b↑∞c↓−∞, d↑∞

∫[a,b)×[c,d)


and hence of the integral∫spec(C)

dE(z)Y F (z) in the uniform operator topology. Si-multaneously, we obtain the estimate (3.3). One can prove the existence of the integral∫spec(C)

F (z)Y ∗ dE(z) in the uniform operator topology and the estimate (3.4) for this

integral by applying Lemma 2.3. �

4. Sylvester equation

This section deals with the operator Sylvester equation (1.8) for the case in which thecoefficient C is a normal operator. The main aim of the section is to construct an explicitrepresentation for the solution X of this equation in the form of an operator Stieltjesintegral with respect to the spectral measure corresponding to C.

First, recall the definitions of weak, strong and operator solutions of the operatorSylvester equation.



Definition 4.1. Let A and C be densely defined closed (possibly unbounded) operatorson Hilbert spaces H and K, respectively. A bounded operator X ∈ B(H,K) is called aweak solution of the Sylvester equation

(4.1) XA− CX = D, D ∈ B(H,K)if

(4.2) 〈XAf, g〉 − 〈Xf,C∗g〉 = 〈Df, g〉 for all f ∈ Dom(A) and g ∈ Dom(C∗).

A bounded operator X ∈ B(H,K) is called a strong solution of the Sylvester equation(4.1) if

(4.3) Ran(X|Dom(A)

)⊂ Dom(C)

and

(4.4) XAf − CXf = Df for all f ∈ Dom(A).

Finally, a bounded operator X ∈ B(H,K) is called an operator solution of the Sylvesterequation (4.1) if

Ran(X) ⊂ Dom(C),

the product XA is a bounded operator on Dom(XA) = Dom(A), and the relation

(4.5) XA− CX = D,

where XA is the closure of XA, holds as an operator identity.

Along with the Sylvester equation (4.1), we consider the dual equation

(4.6) Y C∗ −A∗Y = D∗.

Clearly, if an operator X ∈ B(K,H) is an operator solution of the Sylvester equation(4.1), then it is simultaneously a strong solution of this equation. In turn, every strongsolution is also a weak solution. In fact, one need not distinguish between strong and weaksolutions, because every weak solution of the Sylvester equation (4.1) is automatically astrong solution.

Lemma 4.2. Let A and C be densely defined closed (possibly unbounded) operators onthe Hilbert spaces H and K, respectively, and let D ∈ B(H,K). If an operator X ∈ B(H,K)is a weak solution of the Sylvester equation (4.1), then X is simultaneously a strongsolution of (4.1).

Proof. To each f ∈ Dom(A), we assign a linear functional lf defined on Dom(lf ) =Dom(C∗) by setting

lf (g) = 〈C∗g,Xf〉 = 〈g,XAf〉 − 〈g,Df〉, g ∈ Dom(C∗).

For every f ∈ Dom(A), the functional lf is bounded,

|lf (g)| = |〈C∗g,Xf〉| ≤ cf‖g‖, g ∈ Dom(C∗),

where cf = ‖XAf‖ + ‖Df‖. The domain Dom(lf ) is dense in K, because it coincideswith the domain Dom(C∗) of the adjoint of the densely defined closed operator C. Thus,lf is densely defined in K. Consequently, Xf ∈ Dom

((C∗)∗

)= Dom(C), which means

that the inclusion (4.3) holds and

〈g,XAf − CXf −Df〉 = 0 for all g ∈ H.

Consequently, (4.4) holds as well, which completes the proof of the lemma. �

It is easily seen that if one of equations (4.1) and (4.6) has a weak (and hence strong)solution, then so does the other equation.



Lemma 4.3. An operator X ∈ B(H,K) is a weak (and hence a strong) solution of theSylvester equation (4.1) if and only if the operator Y = −X∗ is a weak (and hence astrong) solution of the dual Sylvester equation (4.6).

Proof. This is a refined (on the basis of Lemma 4.2) version of [4, Lemma 2.4]. �

It is well known that if the spectra of A and C are disjoint and at least one of theseoperators is bounded, then the Sylvester equation XA − CX = D necessarily has asolution, and the solution is unique. M. G. Krein was the first to prove this assertionin 1948. Later, this fact was independently established by Yu. L. Daletskiı [12] andM. Rosenblum [29]. The precise statement is as follows.

Lemma 4.4. Let A be a densely defined closed (possibly unbounded) operator on a Hilbertspace H, and let C be a bounded operator on a Hilbert space K such that

spec(A) ∩ spec(C) = ∅.Next, let D ∈ B(H,K). Then the Sylvester equation (4.1) has a unique operator solution,which has the form

(4.7) X =1

2πi

∫Γ

dζ (C − ζ)−1D(A− ζ)−1,

where the contour Γ is a union of closed contours in the complex plane with total winding0 around every point in spec(A) and total winding 1 around every point in spec(C).Moreover,

‖X‖ ≤ (2π)−1|Γ| supζ∈Γ

‖(C − ζ)−1‖ ‖(A− ζ)−1‖ ‖D‖,

where |Γ| is the length of Γ.

A fairly recent survey of results concerning the operator Sylvester equation (4.1) withbounded coefficients A and C can be found in [6], where applications of these results tovarious specific problems are discussed as well.

It is known that if both A and C are unbounded densely defined closed operators,then the Sylvester equation (4.1) may have no bounded solutions at all even for the casein which the spectra of A and C are separated from each other. (There is a relevantexample in [28].) Nevertheless, (4.1) with unbounded A and C remains solvable undersome additional assumptions. A survey of known sufficient conditions for the solvabilityof (4.1) for the case in which both operators A and C may be unbounded is given in [4,Sec. 2].

Now we shall prove the main result of this section, which states that if one of theoperators A and C is normal, then the strong solution of (4.1) (if it exists) can berepresented by an operator Stieltjes integral. The corresponding representation replacesformula (4.7) if the contour integration around the spectrum of the unbounded operatorC is impossible.

Theorem 4.5. Let A be a densely defined closed (possibly unbounded) operator on aHilbert space H, and let C be a normal (possibly unbounded) operator on a Hilbert spaceK with spectral family {EC(Ω)}Ω∈ABorel(C). Assume that the spectra of A and C aredisjoint, i.e.,

(4.8) dist(spec(A), spec(C)

)> 0.

Let D ∈ B(H,K). Then the following assertions hold.(i) The Sylvester equation (4.1) has at most one strong solution X ∈ B(H,K). If

X ∈ B(H,K) is a strong solution of (4.1), then it can be represented by the operator



Sylvester integral,

(4.9) X =

∫spec(C)

dEC(ζ)D(A− ζ)−1,

which exists in the sense of the strong operator topology on B(H,K).Conversely, if the Stieltjes integral on the right-hand side in (4.9) exists in the sense

of the strong operator topology, then the operator X given by the expression (4.9) is astrong solution of (4.1).

(ii) The dual Sylvester equation

(4.10) Y C∗ −A∗Y = D∗

has at most one strong solution Y ∈ B(K,H). If Y ∈ B(K,H) is a strong solution of(4.10), then it has a representation in the form of the operator Stieltjes integral

(4.11) Y = −∫spec(C∗)

(A∗ − ζ)−1D∗EC∗(dζ),

which exists in the sense of the strong operator topology on B(K,H).The converse is true as well : if the Stieltjes integral on the right-hand side in (4.11)

exists in the sense of the strong operator topology, then the operator Y given by (4.11) isa strong solution of (4.10).

Proof. (i) Set Δ = [a, b)× [c, d), where [a, b) and [c, d) are finite real intervals. Let {δj}be a finite system of pairwise disjoint real intervals such that

⋃j δj = [a, b), and let {ωk}

be another finite system of pairwise disjoint real intervals such that⋃

k ωk = [c, d). Weintroduce the partition of Δ into the smaller triangles Δjk = δj ×ωk. If Δjk ∩σ(C) �= ∅,then to the pair j, k we assign a complex number ζΔjk

∈ C such that the corresponding

real point (Re ζΔjk, Im ζΔjk

) ∈ R2 belongs to the intersection Δjk ∩ σ(C). Let us applythe spectral projection EC(Δjk) to both parts of (4.4). After simple transformations ofthe resulting relation, we find that

(4.12) EC(Δjk)XAf − ζΔjkEC(Δjk)Xf = EC(Δjk)Df + EC(Δjk)(C − ζΔjk

)Xf,

where f ∈ Dom(A) is an arbitrary element. Since (Re ζΔjk, Im ζΔjk

) ∈ Δjk ∩ σ(C), itfollows from (4.8) that ζΔjk

belongs to the resolvent set of A. Hence (4.12) means that

(4.13) EC(Δjk)X = EC(Δjk)D(A− ζΔjk)−1 + (C − ζΔjk

)EC(Δjk)X(A− ζΔjk)−1.

By using (4.13), we find that(4.14) ∑

j,k : Δjk∩σ(C) =∅EC(Δjk)X =

∑j,k : Δjk∩σ(C) =∅

EC(Δjk)D(A− ζΔjk)−1

+∑

j,k : Δjk∩ σ(C) =∅(C − ζΔjk

)EC(Δjk)X(A− ζΔjk)−1.

The left-hand side of (4.14) can be computed in closed form,

(4.15)∑

Δjk∩ σ(C) =∅EC(Δjk)X = EC

(Δ ∩ σ(C)

)X = EC

(Δ)X = EC(Ω)X,

where the set

(4.16) Ω = {ζ ∈ C | a ≤ Re ζ < b, c ≤ Im ζ < d} = [a, b)× i[c, d)

is obtained by the natural embedding of the rectangle Δ in the complex plane C.Obviously, the first term on the right-hand side in (4.14) is an integral sum for the

Stieltjes integral (4.9). More precisely, since the resolvent (A− ζ)−1 is analytic in some



complex neighborhood of the set Ω∩ spec(C), it follows from Theorem 2.5 and definitions(2.53) and (2.55) that

(4.17)

n-limmaxj,k(|δj |+|ωk|)→0

∑j,k : Δjk∩ σ(C) =∅

EC(Δjk)D(A− ζΔjk)−1

=

∫spec(C)∩Ω

dEC(ζ)D(A− ζ)−1.

Using the same argument when considering the second term on the right-hand side in(4.14), we establish that

(4.18)


∑j,k : Δjk∩σ(C) =∅

(C − ζΔjk)EC(Δjk)X(A− ζΔjk

)−1

= C

∫spec(C)∩Ω

dEC(ζ)X(A− ζ)−1 −∫spec(C)∩Ω

dEC(ζ) ζ X(A− ζ)−1.

By using Lemma 2.10, we can readily prove that the right-hand side of (4.18) is justzero, i.e.,


∑j,k : Δjk∩σ(C) =∅

(C − ζΔjk)EC(Δjk)X(A− ζΔjk

)−1 = 0.(4.19)

By passing to the limit as maxj,k(|δj | + |ωk|) → 0 in (4.14) with regard for (4.15),(4.17), and (4.19), we conclude that

(4.20) EC(Ω)X =

∫spec(C)∩Ω

dEC(ζ)D(A− ζ)−1

on every finite triangle Ω of the form (4.16). Since

s-lima, c → −∞b, d → +∞

EC

([a, b)× i[c, d)

)X = X,

we see that (4.20) implies (4.9), which in particular proves the uniqueness of a strongsolution of the operator Sylvester equation (4.1).

To prove the converse claim in (i), assume that the set Ω is defined in (4.16) and theStieltjes integral over Ω on the right-hand side in (4.20) converges in the strong operatortopology as a, c → −∞ and b, d → +∞. We denote the resulting (improper) integral byX. Equation (4.20) with this X holds for every finite triangle Ω of the form (4.16). Itfollows that

CEC(Ω)Xf − EC(Ω)XAf =

∫spec(C)∩Ω

dEC(ζ)D(A− ζ)−1(ζ −A)f

= −∫spec(C)∩Ω

dEC(ζ)Df = −EC(Ω)Df

for each f ∈ Dom(A). Hence

(4.21) CEC(Ω)Xf = EC(Ω)XAf − EC(Ω)Df = EC(Ω)(XAf −Df)

for all f ∈ Dom(A). In particular, it follows from (4.21) that the element CEC(Ω)Xf ,where Ω is defined in (4.16), converges to XAf − Df as a, c → −∞ and b, d → +∞.Since

∫spec(C)∩Ω

ζdEC(ζ)Xf = CEC(Ω)Xf , it also follows from (4.21) that∫spec(C)∩Ω

|ζ|2 d〈EC(ζ)Xf,Xf〉 = supΩ∈P

∥∥EC(Ω)(XAf −Df)∥∥2 = ‖XAf −Df‖2 < ∞,



where P is the set of all rectangles of the form (4.16) in C. Hence

(4.22) Xf ∈ Dom(C).

Then we can permute the operators C and EC(Ω) in the leftmost term in (4.21), thusrewriting (4.21) in the form

(4.23) EC(Ω)CXf = EC(Ω)(XAf −Df)

for all Ω ∈ P. Relations (4.22) and (4.23) together prove that X is a strong solution ofthe Sylvester equation (4.1).

(ii) Assume that the dual Sylvester equation (4.6) has a strong solution Y ∈ B(K,H).Take a rectangle Δ ⊂ R2 having a nonempty intersection with σ(C∗); i.e., assume thatΔ ∩ σ(C∗) �= ∅. Since EC∗(Δ)K ⊂ Dom(C∗), we have Y EC∗(Δ)f ∈ Dom(A∗) for everyf ∈ K by the definition of a strong solution. Take a point ζΔ ∈ spec(C∗) such that(Re ζΔ, Im ζΔ) ∈ Δ. Then it follows from (4.8) that ζΔ �∈ spec(A∗). Just as in the proofof (i), one can readily verify the relation

(4.24) Y EC∗(Δ) = −(A∗ − ζΔ)−1D∗EC∗(Δ)− (A∗ − ζΔ)

−1Y (C∗ − ζΔ)EC∗(Δ),

which is similar to (4.13).Next, assume that [a, b) is a finite real interval and {δj} is a partition of [a, b) into

finitely many pairwise disjoint intervals, [a, b) =⋃

j δj . Likewise, let {ωk} be a finite

system of pairwise disjoint intervals that is a partition of some finite real interval [c, d),⋃k ωk = [c, d). Set Δjk = δj × ωk. If the indexing pair (j, k) satisfies the condi-

tion Δjk ∩ σ(C∗) �= ∅, then to this pair we assign a point ζΔjk∈ spec(C∗) such that

(Re ζΔjk, Im ζΔjk

) ∈ Δjk. In this case, in view of (4.24), we see that

Y EC∗([a, b)× [c, d)

)f = −

∑j,k : Δjk∩ spec(C∗) =∅

(A∗ − ζΔjk)−1D∗EC∗(Δjk)f

(4.25)

−∑

j,k : Δjk∩ spec(C∗) =∅(A∗ − ζΔjk

)−1Y (C∗ − ζΔjk)EC∗(Δjk)f

for each f ∈ K. Meanwhile, (4.19) implies that

(4.26) n-limmaxj,k(|δj |+|ωk|)→0

∑j,k : Δjk∩ spec(C∗) =∅

(A∗ − ζΔjk)−1Y (C∗ − ζΔjk

)EC∗(Δjk) = 0.

Thus, by passing to the limit as maxj,k(|δj |+ |ωk|) → 0 in (4.25), we obtain

(4.27) −∫spec(C∗)∩Ω

(A∗ − ζ)−1D∗dEC∗(ζ)f = Y EC∗([a, b)× [c, d)

)f,

where Ω is defined in (4.16). Since

lima,c→−∞b,d→+∞

Y EC∗([a, b)× [c, d)

)f = Y

for every f ∈ K, we see that the integral on the right-hand side in (4.27) converges in thestrong operator topology as a, c → −∞ and b, d → +∞; moreover, one has (4.11). Thus,a strong solution Y of the dual Sylvester equation (4.10) can necessarily be representedin the form of the strongly convergent integral (4.11). Moreover, this implies that thestrong solution of (4.10) is unique.

To prove the converse claim in (ii), assume that there exists a strong limit

(4.28) Y = s-lima,c→−∞b,d→+∞

∫spec(C∗)∩([a,b)×i[c,d))

(A∗ − ζ)−1D∗dEC∗(ζ), Y ∈ B(K,H).



Then

(4.29) Y EC∗([a, b)× i[c, d)

)= −


(A∗ − ζ)−1D∗dEC∗(ζ)

for any finite a, b, c, and d such that a < b and c < d. By (4.8), every point ξ ∈ spec(C∗)lies in the resolvent set of A∗. We take some ξ ∈ spec(C∗) and represent the operator(4.29) as a sum of two terms,

(4.30) Y EC∗([a, b)× [c, d)

)= J1(a, b, c, d) + J2(a, b, c, d),

where

J1(a, b, c, d) = −(A∗ − ξ)−1D∗EC∗([a, b)× i[c, d)),(4.31)

J2(a, b, c, d) = (A∗ − ξ)−1


(ξ − ζ)(A∗ − ζ)−1D∗dEC∗(ζ).(4.32)

Obviously, (4.32) means that

J2(a, b, c, d)f = −(A∗ − ξ)−1

(∫spec(C∗)∩([a,b)×i[c,d))

(A∗ − ζ)−1D∗dEC∗(ζ)

)(C∗ − ξ)f

for every f ∈ Dom(C∗). Thus,

Y f = lima,c→−∞b,d→+∞

Y EC∗([a, b)× i[c, d)

)f

= lima,c→−∞b,d→+∞

J1(a, b, c, d)f + lima,c→−∞b,d→+∞

J2(a, b, c, d)f

= −(A∗ − ξ)−1D∗f − (A∗ − ξ)−1

(∫spec(C∗)

(A∗ − ζ)−1D∗dEC∗(ζ)

)(C∗ − ξ)f

for every f ∈ Dom(C∗). In other words,

(4.33) Y f = −(A∗ − ξ)−1D∗f + (A∗ − ξ)−1Y (C∗ − ξ)f, f ∈ Dom(C∗),

because

(4.34)

∫spec(C∗)

(A∗ − ζ)−1D∗dEC∗(ζ)

= s-lima,c→−∞b,d→+∞


(A∗ − ζ)−1D∗dEC∗(ζ) = Y

by (4.28). It follows from (4.33) that Y f ∈ Dom(A∗) for every f ∈ Dom(C∗) and hence

(4.35) Ran(Y |Dom(C∗)) ⊂ Dom(A∗).

To verify that Y is a strong solution of (4.10), it remains to apply the operator A∗ − ξto both sides of (4.33). �

Remark 4.6. Under the assumptions of Theorem 4.5, the integral (4.9) converges in thesense of the strong operator topology if and only if so does the integral (4.11). One cansee this by combining Theorem 4.5 with Lemma 4.3. The operators X and Y given bythe integrals (4.9) and (4.11) (if these exist in the sense of the strong operator topology)are related by Y = −X∗.

Lemma 4.7. Let the assumptions of Theorem 4.5 be satisfied. If one of the integrals(4.9) and (4.11) converges in the weak operator topology, then both integrals converge inthe strong operator topology.



Proof. Assume that the integral (4.9) converges in the sense of the weak operator topol-ogy. Set

XΩ =

∫spec(C)∩Ω

dEC(ζ)D(A− ζ)−1,

where Ω = [a, b)× i[c, d) is a finite rectangle in C with −∞ < a < b < ∞ and −∞ < c <d < ∞. According to Remark 2.8, Ran(XΩ) ⊂ Ran

(EC(Ω)

), and hence XΩf ∈ Dom(C).

Thus, by using the same argument as in the proof of (4.21), we obtain

(4.36) XΩAf − CXΩf = EC(Ω)Df

for every f ∈ Dom(A). Relation (4.36) means that

(4.37) 〈XΩAf, g〉 − 〈XΩf, C∗g〉 = 〈EC(Ω)Df, g〉

for all f ∈ Dom(A) and g ∈ Dom(C∗). By the assumptions of the lemma, w-limXΩ = X.At the same time, s-limEC(Ω) = I. Thus, the passage to the limit as a, c → −∞ andb, d → +∞ in (4.37) readily shows that the operator X given by the (improper) weakintegral (4.9) is a weak solution of the Sylvester equation (4.1). But then X is also astrong solution of (4.1) by Lemma 4.2. Further, by applying Theorem 4.5(ii), we concludethat the integral (4.9) converges in the strong operator topology. In view of Remark 4.6,this implies the strong convergence of the operator Stieltjes integral (4.11) as well.

The proof of the desired assertion for the case in which we start from the assumptionthat the integral (4.11) weakly converges can be carried out in a completely similarway. �

The following claim, which generalizes Lemma 2.18 in [4], describes sufficient con-ditions for the existence of a strong (and even an operator) solution of the operatorSylvester equation.

Lemma 4.8. Let the assumptions of Theorem 4.5 be satisfied. Further, suppose that

(4.38) supζ∈ spec(C)

‖(A− ζ)−1‖ < ∞

and that the operator D has finite EC-norm, i.e.,

(4.39) ‖D‖EC< ∞.

Then each of (4.1) and (4.6) has a unique strong solution. The solution X ∈ B(H,K) ofthe first equation has the form (4.9), and the solution Y ∈ B(K,H) of the second equationhas the form (4.11). The Stieltjes integrals in (4.9) and (4.11) converge in the uniformoperator topology, and Y = −X∗.

If, in addition,


‖ζ (A− ζ)−1‖ < ∞,

then

Ran(X) ⊂ Dom(C),(4.41)

Ran(Y ) ⊂ Dom(A∗).(4.42)

In this case, X and Y are operator solutions of (4.1) and (4.6), respectively.

We omit the proof of this lemma, because it coincides with that of Lemma 2.18 in[4] virtually word for word. The only difference is that the operator Stieltjes integralsover the real line are replaced by the corresponding operator Stieltjes integrals over thecomplex plane.



Remark 4.9. Assume that

δ = dist(W(A), spec(C)

)> 0,

where W(A) is the numerical range of A. By [15, Lemma V.6.1], in this case the repre-sentation (4.9) and condition (4.39) imply the estimate

(4.43) ‖X‖EC≤ 1

δ‖D‖EC

.

Remark 4.10. If A is a normal operator, then condition (4.39) with regard for the repre-sentation (4.9) implies the estimate

(4.44) ‖X‖EC≤ 1

d‖D‖EC

,

where d = dist(spec(A), spec(C)

). In this case, the operator X can be represented as a

double operator Stieltjes integral [8],

X =

∫spec(C)

∫spec(A)

dEC(ζ)D

z − ζdEA(z).

If, in addition, D ∈ B2(H,K), then, by [9, Theorem 1], the solution X is a Hilbert–Schmidt operator and satisfies the estimate

‖X‖2 ≤ 1

d‖D‖2.

This estimate is optimal for the Hilbert–Schmidt class B2(H,K).

5. Riccati equation

There exist at least three distinct approaches to studying Riccati equations that con-tain operators on infinite-dimensional spaces. The first approach goes back to Davis [13]and Halmos [16] and is based on the deep relationship between the theory of Riccatiequations and results on the problem concerning perturbations of invariant subspaces ofa linear operator. A detailed discussion of the results obtained in the framework of thisessentially purely geometric approach can be found in the recent paper [18]. We pointout that the capabilities of the geometric approach are largely confined to the case inwhich the Riccati equation is associated with a self-adjoint operator 2 × 2 matrix (e.g.,see [4, Sec. 5]), that is, to the case in which the operators occurring in (1.11) satisfyA = A∗, C = C∗, and D = B∗. However, it is remarkable that the optimal estimatesestablished in [14, 20, 27] (see also [5]) for the rotation of spectral subspaces under off-diagonal perturbations imply the corresponding optimal estimates for the norm of thesolutions of the associated Riccati equations.

The second approach to the analysis of operator Riccati equations is based on thefactorization theorems proved by Markus–Matsaev [22] and Virozub–Matsaev [30] forholomorphic operator functions. This approach has produced a number of results con-cerning sufficient conditions for the solvability of Riccati equations (see [19, 24]), inparticular, for the case in which the operators A and C in (1.11) are not self-adjoint [21].

The third approach [4, 23, 25] (see also [2] and [17]) is closely related to the represen-tation (4.9) of solutions of operator Sylvester equations in the form of operator Stieltjesintegrals with respect to a spectral measure. This representation permits one to reducea Riccati equation to an equivalent integral equation falling within the scope of the Ba-nach fixed point theorem. However, so far this approach has only permitted studyingRiccati equations (1.11) in which at least one of the coefficients A and C is a self-adjointoperator. In this section, we establish corollaries of the integral representation (4.9) for



the more general case of the Riccati equations (1.11) in which one of the coefficients Aand C is a normal operator.

First, recall the definitions of weak, strong, and operator solutions of the Riccatiequation.

Definition 5.1. Let A and C be densely defined closed (possibly unbounded) operatorson Hilbert spaces H and K, respectively. Let B and D be bounded operators from K

to H and from H to K, respectively. A bounded operator X ∈ B(H,K) is called a weaksolution of the Riccati equation

(5.1) XA− CX +XBX = D

if

〈XAf, g〉 − 〈Xf,C∗g〉+ 〈XBXf, g〉 = 〈Df, g〉for any f ∈ Dom(A) and g ∈ Dom(C∗). A bounded operator X ∈ B(H,K) is called astrong solution of the Riccati equation (5.1) if

(5.2) Ran(X|Dom(A)

)⊂ Dom(C)

and

(5.3) XAf − CXf +XBXf = Df

for every f ∈ Dom(A). Finally, X ∈ B(H,K) is called an operator solution of the Riccatiequation (5.1) if

Ran(X) ⊂ Dom(C),

the product XA is a bounded operator on Dom(XA) = Dom(A), and the relation

(5.4) XA− CX +XBX = D,

where XA is the closure of XA, holds as an operator identity.

Along with the Riccati equation (5.1), consider the dual equation

(5.5) Y C∗ −A∗Y + Y B∗Y = D∗.

The following assertion is a corollary of Lemma 4.2.

Lemma 5.2. Let A and C be densely defined closed (possibly unbounded) operatorsin Hilbert spaces H and K, respectively, and let B ∈ B(K,H) and D ∈ B(H,K). IfX ∈ B(K,H) is a weak solution of the operator equation (5.1), then X is also a strongsolution of (5.1).

Proof. The assumption that X is a weak solution of the Riccati equation (5.1) impliesthat this operator is a weak solution of the Sylvester equation

(5.6) XA− CX = D,

where

(5.7) A = A+BX, Dom(A) = Dom(A).

The operator A is closed and densely defined in H. Consequently, by Lemma 4.2, theweak solution X is automatically a strong solution of (5.6), i.e.,

Ran(X|Dom( ˜A)) ⊂ Dom(C) and XAf − CXf = Df for each f ∈ Dom(A).

In view of (5.7), we conclude that X is a strong solution of (5.1). �

Now let us present an assertion that is a straightforward consequence of Lemma 4.3.



Lemma 5.3. Under the assumptions of Lemma 5.2, an operator X ∈ B(H,K) is a weak(and hence a strong) solution of the Riccati equation (5.1) if and only if Y = −X∗ is aweak (and hence a strong) solution of the dual Riccati equation (5.5).

Throughout the remaining part of the paper, we make the following assumption.

Assumption 5.4. Let A be a closed densely defined (possibly unbounded) operator on aHilbert space H, and let C be a normal (possibly unbounded) operator on a Hilbert spaceK. Next, let B ∈ B(K,H) and D ∈ B(H,K).

The theorems in Section 4 concerning the integral representation of solutions of op-erator Sylvester equations permit one to rewrite the Riccati equations (5.1) and (5.5)in the form of equivalent integral equations, whose solvability can be established on thebasis of fixed point theorems.

Theorem 5.5. Let Assumption 5.4 be satisfied. Then(i) An operator X ∈ B(H,K) satisfying the condition

(5.8) dist(spec(A+BX), spec(C)

)> 0

is a weak (and hence a strong) solution of the Riccati equation (5.1) if and only if it isa solution of the integral equation

(5.9) X =

∫spec(C)

dEC(ζ)D (A+BX − ζ)−1,

where the operator Stieltjes integral converges in the weak (and hence in the strong)operator topology.

(ii) An operator Y ∈ B(K,H) satisfying the condition

(5.10) dist(spec(A∗ − Y B∗), spec(C∗)

)> 0

is a weak (and hence a strong) solution of the dual Riccati equation (5.5) if and only ifit is a solution of the integral equation

(5.11) Y = −∫spec(C∗)

(A∗ − Y B∗ − ζ)−1D∗ dEC∗(ζ),

where the operator Stieltjes integral converges in the weak (and hence in the strong)operator topology.

Proof. (i) An operator X is a weak (and hence a strong) solution of (5.1) if and only ifit is a weak (and hence a strong) solution of the Sylvester equation

XA− CX = D,

where

A = A+BX.

Hence the proof of (i) amounts to a reference to Theorem 4.5 (i) and Lemma 4.7.(ii) An operator Y is a weak (and hence a strong) solution of (5.5) if and only if it is

a weak (and hence a strong) solution of the Sylvester equation

Y C − AY = D∗,

where

A = A− Y B∗.

Hence the proof of (ii) amounts to a reference to Theorem 4.5 (ii) and Lemma 4.7. �

The following assertion is a straightforward consequence of Lemma 4.8.



Theorem 5.6. Let Assumption 5.4 be satisfied. Assume also that the operator D hasfinite EC-norm,

(5.12) ‖D‖EC< ∞,

and the Riccati equation (5.1) has a weak solution X ∈ B(H,K) such that

(5.13) dist(spec(A+BX), spec(C)

)> 0

and


‖(A+BX − ζ)−1‖ < ∞.

Then X is also a strong solution of (5.1), and Y = −X∗ is a strong solution of thedual Riccati equation (5.5). The solutions X and Y can be represented as the operatorStieltjes integrals

X =

∫spec(C)

dEC(ζ)D (A+BX − ζ)−1,(5.15)

Y = −∫spec(C)

(A− Y B∗ − ζ)−1D∗dEC∗(ζ)(5.16)

convergent in the uniform operator topology. Moreover, X and Y have finite EC-norm,which satisfies the estimate

(5.17) ‖Y ‖EC= ‖X‖EC

≤ ‖D‖ECsup

ζ∈spec(C)

‖(A+BX − ζ)−1‖.

If condition (5.14) is replaced by the condition


‖ζ (A+BX − ζ)−1‖ < ∞,

then

Ran(X) ⊂ Dom(C), Ran(Y ) ⊂ Dom(A∗)

and hence the strong solutions X and Y are operator solutions of the Riccati equations(5.1) and (5.5), respectively.

If the spectrum of the normal operator C is separated from the numerical range W(A)of the operatorA, then we can prove the existence of a fixed point of the mapping specifiedby the right-hand side of (5.9). Needless to say, a certain smallness of the operators Band D should be assumed. If the operator A is normal as well, then there exists a fixedpoint under weaker assumptions.

Theorem 5.7. Let Assumption 5.4 be satisfied, and let B �= 0. Next, assume that oneof the following conditions is satisfied :

(i) d = dist(W(A), spec(C)

)> 0.(5.19)

(ii) The operator A is normal, and d = dist(spec(A), spec(C)

)> 0.(5.20)

Furthermore, assume that D has finite EC-norm and that

(5.21)√

‖B‖ ‖D‖EC<

d

2.

Then the Riccati equation (5.1) has a strong solution X ∈ B(H,K) whose EC-norm isfinite,

(5.22) ‖X‖EC≤ 1

‖B‖

(d

2−√

d2

4− ‖B‖ ‖D‖EC

).



Moreover, X is the unique strong solution of (5.1) in the operator ball

(5.23){T ∈ B(H,K) | ‖T‖ < ‖B‖−1

(d−

√‖B‖ ‖D‖EC

)}.

If, moreover,

(5.24) ‖B‖ <d

2and ‖B‖+ ‖D‖EC

< d

or

(5.25) ‖B‖ ≥ d

2

(and hence ‖D‖EC

<d2

4‖B‖),

then

(5.26) ‖X‖ ≤ ‖X‖EC< 1;

i.e., the strong solution X is a strict contraction both in the uniform operator topologyand in the topology generated by the EC-norm.

Proof. The proof of both (i) and (ii) reproduces that of the second part of [4, Theorem3.6] almost word for word. The only difference is that in case (i), instead of the exactformula

(5.27) ‖(A− ζ)−1‖ =1

dist(ζ, spec(A)

) , ζ �∈ spec(A),

for the norm of the resolvent, which holds if A is a normal (e.g., self-adjoint) operator,one should use the estimate

‖(A− ζ)−1‖ ≤ 1

dist(ζ,W(A)), ζ �∈ W(A).

Note that this estimate is the main assertion of [15, Lemma V.6.1]. �

Remark 5.8. Theorem 5.7 generalizes the results obtained in [4, 23, 25, 26] for the casein which both operators A and C are self-adjoint. The solvability of the equation (5.15)under condition (5.20) was also studied in [2], where the case was considered in which A isa densely defined closed operator (not necessarily self-adjoint and possibly unbounded), Bis a bounded operator, C is a bounded self-adjoint operator, and D is a Hilbert–Schmidtoperator.

References

[1] V. M. Adamyan, H. Langer, R. Mennicken, and J. Saurer, Spectral components of selfadjoint blockoperator matrices with unbounded entries, Math. Nachr. 178 (1996), 43–80. MR1380703 (97i:47036)

[2] V. Adamyan, H. Langer, and C. Tretter, Existence and uniqueness of contractive solutions of someRiccati equations, J. Funct. Anal. 179 (2001), no. 2, 448–473. MR1809118 (2001j:34074)

[3] N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover, New York,1993. MR1255973 (94i:47001)

[4] S. Albeverio, K. A. Makarov, and A. K. Motovilov, Graph subspaces and the spectral shift func-tion, Can. J. Math. 55 (2003), no. 3. 449–503; electronic version: arXiv:math.SP/0105142 v3.MR1980611 (2004d:47031)

[5] S. Albeverio, A. K. Motovilov, and A. V. Selin, The a priori tan θ theorem for eigenvectors,SIAM J. Matrix Anal. Appl. 29 (2007), no. 2, 685–697; electronic version: arXiv:math.SP/0512545.MR2318372 (2008b:47039)

[6] R. Bhatia and P. Rosenthal, How and why to solve the operator equation AX − XB = Y , Bull.London Math. Soc. 29 (1997), no. 1, 1–21. MR1416400 (97k:47016)

[7] M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators on a Hilbert Space,Izdat. Leningrad. Univ., Leningrad, 1980; English transl., Kluwer, Dordrecht, 1987. MR609148(82k:47001)

[8] M. Sh. Birman and M. Z. Solomyak, Double operator integrals in a Hilbert space, Integr. Eq. Oper.Th. 47 (2003), no. 2, 131–168. MR2002663 (2004f:47029)


http://www.ams.org/mathscinet-getitem?mr=1380703

















[9] M. Sh. Birman and M. Z. Solomyak, Stieltjes double-operator integrals, in Topics in mathematicalphysics, vol. 1, Consultants Bureau, New York, 1967, pp. 25–54. MR0209872 (35:767b)

[10] M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals. II, in Topics in mathematicalphysics, vol. 2, Consultants Bureau, New York, 1968, pp. 19–46. MR0234304 (38:2621)

[11] M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals. III, in Problems of mathe-matical physics. No. 6. Theory of functions. Spectral theory. Wave propagation, Izdat. Leningrad.Univ., Leningrad, 1973, pp. 27–53. (Russian) MR0348494 (50:992)

[12] Yu. L. Daletskiı, On the asymptotic solution of a vector differential equation, Dokl. Akad. NaukSSSR 92 (1953), no. 5, 881–884. MR0060094 (15:624h)

[13] C. Davis, Separation of two linear subspaces, Acta Scient. Math. (Szeged) 19 (1958), 172–187.MR0098980 (20:5425)

[14] C. Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer.Anal. 7 (1970), 1–46. MR0264450 (41:9044)

[15] I. Ts. Gohberg and M. G. Kreın, Introduction to the theory of linear non-selfadjoint operators inHilbert space, Nauka, Moscow, 1965; English transl., Transl. of Math. Monographs, vol. 18, Amer.Math. Soc., Providence, RI, 1969. MR0220070 (36:3137)

[16] P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389. MR0251519 (40:4746)

[17] V. Hardt, R. Mennicken, and A. K. Motovilov, Factorization theorem for the transfer functionassociated with a 2× 2 operator matrix having unbounded couplings, J. Oper. Th. 48 (2002), no. 1,187–226. MR1926050 (2003g:47026)

[18] V. Kostrykin, K. A. Makarov, and A. K. Motovilov, Existence and uniqueness of solutions to theoperator Riccati equation. A geometric approach, in Advances in differential equations and mathe-matical physics (Birmingham, AL, 2002), Contemp. Math., vol. 327, Amer. Math. Soc., Providence,RI, 2003, pp. 181–198; electronic version: arXiv:math.SP/0207125. MR1991541 (2004f:47012)

[19] V. Kostrykin, K. A. Makarov, and A. K. Motovilov, On the existence of solutions to the operatorRiccati equation and the tanΘ theorem, Integr. Eq. Oper. Th. 51 (2005), no. 1, 121–140; electronicversion: arXiv:math.SP/0210032 v2. MR2120094 (2005j:47070)

[20] V. Kostrykin, K. A. Makarov, and A. K. Motovilov, Perturbation of spectra and spectral subspaces,Trans. Amer. Math. Soc. 359 (2007), no. 1, 77–89; electronic version: arXiv:math.SP/0306025.MR2247883 (2008b:47026)

[21] H. Langer, A. Markus, V. Matsaev, and C. Tretter, A new concept for block operator matrices:the quadratic numerical range, Linear Algebra Appl. 330 (2001), no. 1–3, 89–112. MR1826651(2002b:47015)

[22] A. Markus and V. Matsaev, On the spectral theory of holomorphic operator-valued functions inHilbert space, Funktsional. Anal. Prilozhen. 9 1975, no. 1, 76–77; English transl., Funct. Anal.Appl. 9 (1975), no. 1, 73–74. MR0370241 (51:6468)

[23] R. Mennicken and A. K. Motovilov, Operator interpretation of resonances arising in spectral prob-lems for 2×2 operator matrices, Math. Nachr. 201 (1999), 117–181; electronic version: arXiv:funct-an/9708001. MR1680916 (2000f:47040)

[24] R. Mennicken and A. A. Shkalikov, Spectral decomposition of symmetric operator matrices, Math.

Nachr. 179 (1996), 259–273. MR1389460 (97d:47012)[25] A. K. Motovilov, Potentials appearing after removal of the energy-dependence and scattering by

them, Proc. of the Intern. Workshop “Mathematical aspects of the scattering theory and applica-tions,” St. Petersburg University, St. Petersburg, 1991, pp. 101–108.

[26] A. K. Motovilov, Removal of the resolvent-like energy dependence from interactions and invariantsubspaces of a total Hamiltonian, J. Math. Phys. 36 (1995), no. 12, 6647–6664; electronic version:arXiv:funct-an/9606002. MR1359649 (96m:81249)

[27] A. K. Motovilov and A V. Selin, Some sharp norm estimates in the subspace perturbationproblem, Integral Equations Operator Theory 56 (2006), no. 4, 511–542; electronic version:arXiv:math.SP/0409558 v2. MR2284713 (2008e:47032)

[28] V. Q. Phong, The operator equation AX−XB = C with unbounded operators A and B and relatedabstract Cauchy problems, Math. Z. 208 (1991), no. 4, 567–588. MR1136476 (93b:47035)









































[29] M. Rosenblum, On the operator equation BX − XA = Q, Duke Math. J. 23 (1956), 263–269.MR0079235 (18:54d)

[30] A. I. Virozub and V. I. Matsaev, The spectral properties of a certain class of self-adjoint operatorfunctions, Funktsional. Anal. Prilozhen. 8 (1974), no. 1, 1–10, English transl., Funct. Anal. Appl.8 (1974), no. 1, 1–9.

Institut fur angewandte Mathematik, Universitat Bonn, Wegelerstraße 6, D-53115 Bonn,

Germany

E-mail address: [email protected]

Joint Institute for Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow Region, Rus-

sian Federation

E-mail address: [email protected]

Translated by V. E. NAZAIKINSKII




operator stieltjes integrals with respect to a spectral ... · in the usual way as the limits (if...

Documents