Grupo de Investigación About the Mathematical Foundation of Quantum Mechanics

M. Victoria Velasco Collado

Departamento de Análisis Matemático

Universidad de Granada (Spain)

Operator Theory and The Principles of Quantum Mechanics

CIMPA-MOROCCO research school,

Meknès, September 8-17, 2014

Lecture nº 2

12-09-2014

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Grupo de Investigación About the Mathematical Foundation of Quantum Mechanics

Operator Theory and The Principles of Quantum Mechanics

CIMPA-MOROCCO MEKNÈS, September 2014

Lecture 1: About the origins of the Quantum Mechanics

Lecture 2: The mathematical foundations of Quantum Mechanics.

Lecture 3: About the future of Quantum Mechanics. Some problems and challenges

Hilbert spaces: orthogonality, summable families, Fourier expansion,

the prototype of Hilbert space, Hilbert basis, the Riesz-Fréchet

theorem, the weak topology.

Operators on Hilbert spaces: The adjoint operator, the spectral

equation, the spectrum, compact operators on Hilbert spaces.

Lecture 2: The mathematical foundations of Quantum Mechanics

From Quantum Mechanics to Functional Analysis

Definition: A Banach space is a complete normed space.

Definition: A normed space is a (real or complex) linear space 𝑋 equipped

with a norm, i. e. a function ∙ : 𝑋 → ℝ satisfying

i) 𝑥 = 0 ⇒ 𝑥 = 0 (separates points)

ii) 𝛼𝑥 = 𝛼 𝑥 (absolute homogeneity)

iii) 𝑥 + 𝑦 ≤ 𝑥 + 𝑦 triangle inequality (or subadditivity).

Definition: A Hilbert space is a inner product Banach space 𝐻. That is a

Banach space 𝐻 whose norm is given by 𝑥 = 𝑥, 𝑥 , for every 𝑥 ∈ 𝐻, where ∙,∙ is an inner product, i. e. a mapping ∙,∙ : 𝐻 × 𝐻 → K such that:

i) 𝑥, 𝑥 = 0 ⇒ 𝑥 = 0

ii) 𝑥, 𝑥 ≥ 0

iii) 𝛼𝑥 + 𝛽𝑦, 𝑧 = 𝛼 𝑥, 𝑧 + 𝛽 𝑦, 𝑧

iv) 𝑥, 𝑦 = 𝑦, 𝑥

Note: If 𝐻 is a real space, then the conjugation is the identity map.

v

Hilbert spaces Examples of Banach spaces and Hilbert spaces:

a) ℝ, ℂ (Hilbert)

b) ℝ𝑛, ℂ𝑛 (Hilbert)

c) Sequences spaces 𝑙𝑝 (Hilbert si 𝑝 = 2)

d) Spaces of continuous functions 𝐶,𝑎, 𝑏- e) Spaces of integrable functions 𝐿𝑝,𝑎, 𝑏- (Hilbert if 𝑝 = 2)

f) Matrices 𝑀𝑛×𝑛 (Hilbert)

g) Spaces of bounded linear operators: 𝐿 𝑋, 𝑌 , 𝐿(𝐻)

Espacios de Hilbert

Espacios de Banach

Theorem (Cauchy-Schawarz inequality): If 𝐻 is a linear space equipped with a inner

product ∙,∙ then

𝑢, 𝑣 ≤ 𝑢, 𝑢 𝑣, 𝑣 (𝑢, 𝑣 ∈ 𝐻)

From the Cauchy-Schawarz inequality we obtain straightforwardly the Minkowsky

inequality:

𝑢 + 𝑣, 𝑢 + 𝑣 ≤ 𝑢, 𝑢 + 𝑣, 𝑣 𝑢, 𝑣 ∈ 𝐻

Theorem: If 𝐻 is a linear space equipped with a inner product ∙,∙ , then 𝐻 is a

normed space with the norm 𝑢 = 𝑢, 𝑢 (𝑢 ∈ 𝐻).

Hilbert spaces An inner product space is linear space 𝐻 equipped with an inner product ∙,∙ .

Let us rewrite Cauchy-Schwarz inequality in terms of the associated norm:

Fact: If 𝐻 is a inner product space, then

𝑢, 𝑣 ≤ 𝑢 𝑣 (𝑢, 𝑣 ∈ 𝐻) Cauchy-Schwarz

Corollary: If 𝐻 is a inner product space, then ∙,∙ is continuous.

Polarization identities: Let 𝐻 be a inner product space over K

If K = ℝ, then 𝑢, 𝑣 =1

4 ( 𝑢 + 𝑣 2 − 𝑢 − 𝑣 2).

If K = ℂ, then 𝑢, 𝑣 =1

4 ( 𝑢 + 𝑣 2 − 𝑢 − 𝑣 2)+

𝑖

4 ( 𝑢 + 𝑖𝑣 2 − 𝑢 − 𝑖𝑣 2).

This means that if 𝑢 = 𝑙𝑖𝑚 𝑢𝑛 and 𝑣 = 𝑙𝑖𝑚 𝑣𝑛, then 𝑢, 𝑣 = lim 𝑢𝑛, v𝑛 .

I fact, in terms of the associated norm, the inner product is given by:

Consequently, if 𝐻 is a inner product space over K (= ℝ or ℂ) then

Parallelogram identity: 𝑢 + 𝑣 2 + 𝑢 − 𝑣 2 = 2( 𝑢 2+ 𝑣 2).

v

Hilbert spaces

Question: Given a normed space 𝐻, how to know if 𝐻 is an inner product space?

If this is the case, then such a norm needs to satisfy the paralelogram indetity

𝑢

𝑣

𝑢 −v

v

Theorem (Parallelogram theorem): If (𝐻, ∙ ) is a normed space then ∙ is given

by an inner product if and only of if, for every 𝑢, 𝑣 ∈ 𝐻 we have that

𝑢 + 𝑣 2 + 𝑢 − 𝑣 2 = 2( 𝑢 2+ 𝑣 2) (Parallelogram identity)

Therefore we know, for instance that:

𝑙𝑝 is Hilbert space ⇔ 𝑝 = 2.

Theorem: Any inner product space may be completed to a Hilbert space.

Proof (sketch): If 𝑢 = 𝑙𝑖𝑚 𝑢𝑛 and 𝑣 = 𝑙𝑖𝑚 𝑣𝑛 (for 𝑢, 𝑣 ∈ 𝐻 ) then define

𝑢, 𝑣 : = 𝑙𝑖𝑚 𝑢𝑛, 𝑣𝑛 .

It happens that this not depend of the choice of the sequences. Moreover,

𝑢 = 𝑙𝑖𝑚 𝑢𝑛 = 𝑙𝑖𝑚 𝑢𝑛, 𝑢𝑛 = 𝑢, 𝑢 If follows that 𝐻 is Hilbert space from the parallelogram identity.

Hilbert spaces: orthogonality

𝑎

𝑣

Law of cosines (Euclides): 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 𝑐𝑜𝑠𝜃

Fact : 𝑢 ⊥ 𝑣 ⟺ 𝑐𝑜𝑠𝜃=0 ⟺ 𝑢, 𝑣 =0

𝑏

𝑐

𝜃

𝜃

𝑣

𝑣

𝑢 − 𝑣

Definition: Let 𝐻 be an inner space. It is said that 𝑢, 𝑣 ∈ 𝐻 are orthogonal vectors

if 𝑢, 𝑣 =0. If this is the case, then we write 𝑢 ⊥ 𝑣.

Definition: Let 𝐻 be an inner space. The orthogonal set of S ⊆ 𝐻 is defined by

𝑆⊥≔ 𝑢 ∈ 𝐻: 𝑢 ⊥ s, ∀𝑠 ∈ 𝑆 .

Proposition: If 𝐻 is an inner space, and if S, W ⊆ 𝐻 then:

i) 0 ∈ 𝑆⊥ iv) 𝑆 ⊆ 𝑊 ⟹ 𝑊⊥ ⊆ 𝑆⊥

ii) 0 ∈ 𝑆 ⟹ 𝑆 ∩ 𝑆⊥ = 0 v) 𝑆⊥ is a closed linear subspace of H iii) *0+⊥= 𝐻; 𝐻⊥ = 0 vi) 𝑆 ⊆ 𝑆⊥⊥

Consequently:

𝑐𝑜𝑠𝜃 =𝑢 2+ 𝑣 2− 𝑢−𝑣 2

2 𝑢 𝑣=

𝑢,𝑢 + 𝑣,𝑣 − 𝑢−𝑣,𝑢−𝑣

2 𝑢 𝑣=

2 𝑢,𝑣

2 𝑢 𝑣 =

𝑢

𝑢,

𝑣

𝑣

Hilbert spaces: orthogonality

Definition: Let 𝑋 be a linear space, and let 𝑌 and 𝑍 be linear subspaces. Then:

𝑋 = 𝑌⨁𝑍 ⟺ 𝑋 = 𝑌 + 𝑍 ; Y ∩ 𝑍 = 0 . (Direct sum)

Let 𝑋 be a normed space such that 𝑋 = 𝑌⨁𝑍. Let 𝑥 = 𝑦 + 𝑧 and 𝑥𝑛 = 𝑦𝑛 + 𝑧𝑛 in 𝑌⨁𝑍.

We hope that : 𝑥𝑛→ 𝑥 ⟺ 𝑦𝑛 → 𝑥 y 𝑧𝑛 → 𝑧. If this is the case, then we say that 𝑋 = 𝑌⨁𝑍 is a topological direct sum.

Theorem (best approximation): Let 𝐻 be a Hilbert space, 𝑢 ∈ 𝐻, and 𝑀 a closed

subspace of 𝐻. Then, there exists a unique 𝑚 ∈ 𝑀 such that 𝑢 − 𝑚 = 𝑑 𝑢, 𝑀 .

Orthogonal projection theorem: Let 𝐻 be Hilbert space and M a closed subspace.

Then, 𝐻 = 𝑀 ⨁ 𝑀⊥. Morever if 𝜋𝑀: 𝐻 → 𝑀 is the canonical projection then 𝜋𝑀 = 𝑃𝑀. Also 𝜋𝑀 = 1

and 𝑢 = 𝜋𝑀(𝑢) + 𝑢 − 𝜋𝑀(𝑢) = 𝑃𝑀(𝑢) + 𝑢 − 𝑃𝑀(𝑢) .

Notation: 𝑃𝑀 𝑢 = 𝑚 ∈ 𝑀: 𝑢 − 𝑚 = 𝑑 𝑢, 𝑀 = best approximation from 𝑢 to 𝑀.

Fact: If 𝑋 is a Banach space then,

𝑋 = 𝑌⨁𝑍 is a topological direct sum ⟺ the subspaces 𝑌 and 𝑍 are closed

Remmark: Results also know as Hilbert projection theorem

(Topological direct sum)

Hilbert spaces: orthogonality

Fact: Every closed subspace of a Hilbert space admits a topological complement.

Theorem (Lindestrauss-Tzafriri, 1971): Assume that every closed subspace of a

Banach space 𝑋 is complemented. Then 𝑋 is isomophic to a Hilbert space.

.

This is a very important property. Indeed it characterizes the Hilbert spaces

To be complemented means to admit a topological complement

Hilbert spaces: orthogonality

Proposición: Every orthogonal family of vectors in 𝐻 is linearly independent.

Definition: Let *𝑒𝑖+𝑖∈𝐼 be a family of nonzero vectors of 𝐻. We say that *𝑒𝑖+𝑖∈𝐼 is a

orthogonal family if 𝑒𝑖 , 𝑒𝑗 = 0 ∀𝑖 ≠ 𝑗.

If in addition 𝑒𝑖 = 1, ∀𝑖 ∈ 𝐼, then we say that *𝑒𝑖+𝑖∈𝐼 is a ortonormal family.

Proposition (Gram–Schmidt process): Let *𝑣𝑛+𝑛∈ℕ be a linearly independent family

of vectors in a inner space 𝐻. Then the family *𝑢𝑛+𝑛∈ℕ given by

𝑢1= 𝑣1

𝑢𝑛= 𝑣𝑛 − 𝑢𝑗 , 𝑣𝑛

𝑢𝑗 , 𝑢𝑗

𝑛−1

𝑗=1

𝑢𝑗

is a orthogonal family in 𝐻.

Morever the family *𝑒𝑛+𝑛∈ℕ given by 𝑒𝑛 =𝑢𝑛

𝑢𝑛 is orthonormal.

The Gram–Schmidt process is a method for orthonormalising a set of vectors in an

inner product space. It applies to a linearly independent countably infinite (or finite)

family of vectors in a inner space.

Hilbert spaces : summable families

Proposition: A family of positive real numbers *𝑥𝑖+𝑖∈𝐼 is summable if and only if the

set * 𝑥𝑖𝑖∈𝐽 : J ⊆ 𝐼 ; 𝐽 𝑓𝑖𝑛𝑖𝑡𝑜+ is bounded. If this is the case then,

𝑥 = 𝑥𝑖

𝑖∈𝐼

= sup* 𝑥𝑖

𝑖∈𝐽

: J ⊆ 𝐼 ; 𝐽 𝑓𝑖𝑛𝑖𝑡𝑜+ .

Remark: Apply the Cauchy condition with 𝜀 =1

𝑛. Then we obtain 𝐽1

𝑛

such that

𝑥𝑖𝑖∈𝐽 < 1

𝑛 if 𝐽 ∩ 𝐽𝜀 = ∅. Hence, 𝐼0 ≔ 𝐽1

𝑛

is a countable set. Moreover if 𝑖 ∈ 𝐼 ∖ 𝐼0

then, 𝑥𝑖 <1

𝑛, ∀𝑛, so that 𝑥𝑖 = 0.

Definition: A family *𝑥𝑖+𝑖∈𝐼 in a normed space 𝑋 is summable if there exists 𝑥 ∈ 𝑋

such that ∀𝜀 > 0 there exists a finite set 𝐽𝜀 ⊆ 𝐼, such that if J ⊆ 𝐼 is finite and

𝐽𝜀 ⊆ 𝐽 then 𝑥𝑖𝑖∈𝐽 − 𝑥 < 𝜀. If this is the case we write 𝑥 = 𝑥𝑖𝑖∈𝐼 .

Definition: A family *𝑥𝑖+𝑖∈𝐼 in a normed space 𝑋 satisfies the Cauchy condition if

∀ 𝜀 > 0 there exists a finite 𝐽𝜀 ⊆ 𝐼, such that if J ⊆ 𝐼 is finite, and if 𝐽 ∩ 𝐽𝜀 = ∅ then,

𝑥𝑖𝑖∈𝐽 < 𝜀.

Theorem: Let 𝑋 be a Banach space and *𝑥𝑖+𝑖∈𝐼 a family in 𝑋. Then,

i) *𝑥𝑖+𝑖∈𝐼 is summable ⟺ *𝑥𝑖+𝑖∈𝐼 satisfies the Cauchy condition.

ii) *𝑥𝑖+𝑖∈𝐼 summable ⟹ 𝑖 ∈ 𝐼: 𝑥𝑖 ≠ 0 is countable.

Hilbert spaces : summable families

Theorem: Let *𝑒𝑖+𝑖∈𝐼 be a orthogonal family *𝑒𝑖+𝑖∈𝐼 in a Hilbert space 𝐻. Then,

*𝑒𝑖+𝑖∈𝐼 summable (in 𝐻) ⟺ * 𝑒𝑖2+𝑖∈𝐼 is summable en ℝ,

in whose case:

The idea is to generalize this fact to an arbitrary 𝐻 to have “coordinates”.

This result addresses the problem of the summability in 𝐻 to the problem of the

summability in ℝ. On the other hand, it generalizes the Pythagoras's theorem.

Remark: The canonical base of ℝ3 is orthogonal. Moreover if 𝑥 ∈ ℝ3 is such that

𝑥 = (𝑥1, 𝑥2, 𝑥3), that is 𝑥 = 𝑥𝑖𝑒𝑖 ,𝑖=3𝑖=1 then

Remark: Let *𝑒𝑖+𝑖∈𝐼 be a othonormal family in a Hilbert space 𝐻. Then

*𝑓(𝑒𝑖)𝑒𝑖+𝑖∈𝐼 summable in 𝐻 ⟺ * 𝑓(𝑒𝑖)2+𝑖∈𝐼 es summable in ℝ .

If this is the case, then: 𝑓(𝑒𝑖)𝑒𝑖𝑖∈𝐼2 = 𝑓(𝑒𝑖)

2𝑖∈𝐼 .

Particularly: * 𝑥, 𝑒𝑖 𝑒𝑖+𝑖∈𝐼 summable in 𝐻 ⟺ * 𝑥, 𝑒𝑖2+𝑖∈𝐼 summable ℝ .

𝑒𝑖𝑖∈𝐼2 = 𝑒𝑖

2𝑖∈𝐼 .

𝑥 = 𝑥, 𝑒𝑖 𝑒𝑖

𝑖=3

𝑖=1

Hilbert spaces : summable families

.

Theorem (Bessel's inequality): If *𝑒𝑖+𝑖∈𝐼 is a orthormal family in a Hilbert space 𝐻, then for every 𝑢 ∈ 𝐻, the family 𝑢, 𝑒𝑖 𝑒𝑖𝑖∈𝐼 is summable. Moreover

𝑢, 𝑒𝑖 𝑒𝑖

𝑖∈𝐼

≤ 𝑢

When * 𝑥, 𝑒𝑖2+𝑖∈𝐼 is summable in ℝ ? When is 𝑥 = 𝑥, 𝑒𝑖 𝑒𝑖𝑖∈𝐼 ?

Let 𝐽 ⊆ 𝐼, such that 𝐽 is finite. Then

0 ≤ 𝑥 − 𝑥, 𝑒𝑖 𝑒𝑖

𝑖∈𝐽

2

= 𝑥 − 𝑥, 𝑒𝑖 𝑒𝑖

𝑖∈𝐽

, 𝑥 − 𝑥, 𝑒𝑖 𝑒𝑖

𝑖∈𝐽

=⋅⋅⋅= 𝑥 2 − 𝑥, 𝑒𝑖2

𝑖∈𝐽

Therefore, 𝑥, 𝑒𝑖2

𝑖∈𝐽 ≤ 𝑥 2 for every 𝐽 ⊆ 𝐼, with 𝐽 finite.

Consequently

𝑢, 𝑒𝑖2

𝑖∈𝐼 = sup* 𝑢, 𝑒𝑖2

𝑖∈𝐽 : J ⊆ 𝐼 ; 𝐽 𝑓𝑖𝑛𝑖𝑡e+ ≤ 𝑥 2

This means that * 𝑥, 𝑒𝑖

2+𝑖∈𝐼 is summable in ℝ and hence we have the following:

Hilbert spaces : summable families

.

Corollary (characterization of the maximal ortonormal families): Let *𝑒𝑖+𝑖∈𝐼 be an

orthonormal system in a Hilbert space 𝐻. The following conditions are equivalent:

i) 𝑢 = 𝑢, 𝑒𝑖 𝑒𝑖𝑖∈𝐼 ii) 𝑢, 𝑣 = 𝑢, 𝑒𝑖 𝑒𝑖 , 𝑣𝑖∈𝐼 (Parseval’s identy)

iii) 𝑢 = 𝑢, 𝑒𝑖2

𝑖∈𝐼

iv) *𝑒𝑖+𝑖∈𝐼 is a maximal orthonormal maximal

v) 𝑢 ⊥ 𝑒𝑖 ∀𝑖 ⇒ 𝑢 = 0

vi) 𝐻 = 𝑙𝑖𝑛 𝑒𝑖: 𝑖 ∈ 𝐼 .

Proof: TPO+Bessel: 𝑢 ∈ 𝐻 = 𝑙𝑖𝑛 𝑒𝑖: 𝑖 ∈ 𝐼 ⟺ 𝑢 = 𝑢, 𝑒𝑖 𝑒𝑖𝑖∈𝐼 . The other assertions are easy to prove.

Fact: By Zorn’s lemma there exist maximal ortonormal basis.

Definition: A Hilbert basis (or an orthonormal basis) in 𝐻 is a maximal orthonormal

family of vectors in 𝐻.

Example: *𝑒𝑛+𝑛∈ℕ is an orthonormal family in 𝑙2 = *𝑢 = 𝛼𝑛 𝑛∈ℕ ∶ 𝛼𝑛2 < ∞+

such that 𝑢 ⊥ 𝑒𝑛 ∀𝑛 ⇒ 𝑢 = 0. Therefore 𝐵 ≔ *𝑒𝑛: 𝑛 ∈ ℕ+ is a Hilbert basis of 𝑙2.

Note that 𝐵 is not a Hamel basis. Inndeed the linear span of 𝐵 is 𝑐00 (sequences

with finite support). Similarly with 𝑙2(I) where 𝑙2 = 𝑙2(ℕ).

Hilbert spaces : Fourier expansion

Definition: The Hilbert space dimension of a Hilbert space 𝐻 is the cardinality of an

orthonormal basis of 𝐻.

Theorem: Every Hilbert space 𝐻 has an orthonormal basis. Moreover all the

orthonormal bases of 𝐻 have have the same cardinal.

Definition: Let 𝐻 be a Hilbert space and *𝑒𝑖+𝑖∈𝐼 a Hilbert basis. For every 𝑢 ∈ 𝐻, 𝑢 = 𝑢, 𝑒𝑖 𝑒𝑖𝑖∈𝐼 (Fourier expansion)

Fact: The Fourier coefficients * 𝑢, 𝑒𝑖 +𝑖∈𝐼 are the unique family of scalars *𝛼𝑖+𝑖∈𝐼

such that 𝑢 = 𝛼𝑖𝑒𝑖𝑖∈𝐼 vector coordinates in ∞-dim!!

Remark: Denote by dim𝐻 the algebraic dimension of 𝐻. If dim𝐻 < ∞ , then 𝐵 = 𝑒1,…, 𝑒𝑛 is a Hilbert basis ⟺ 𝐵 is a Hamel basis.

Indeed,

𝐻 = 𝑙𝑖𝑛*𝑒1,…, 𝑒𝑛+ = 𝑙𝑖𝑛*𝑒1,…, 𝑒𝑛+.

If dim𝐻 = ∞, and if 𝐵 = *𝑒𝑖+𝑖∈𝐼 is a Hilbert basis then we have 𝐻 = 𝑙𝑖𝑛*𝑒𝑖: 𝑖 ∈ 𝐼+ . But *𝑒𝑖+𝑖∈𝐼 do not need to be a spanning set of 𝐻. Therefore

Hilbert basis ⇏ Hamel basis

Hilbert spaces : Fourier expansion

Example: Let be the Lebesgue 𝜎-algebra of −𝜋, 𝜋 and let 𝑚 be the

Lebesgue measure. Let 𝜇 =𝑚

𝜋 if K = ℝ, and 𝜇 =

𝑚

2𝜋 if K = ℂ . Then

The family 𝑒𝑖𝑛𝑡: 𝑛 ∈ ℤ is a Hilbert basis of 𝐿2( −𝜋, 𝜋 , , 𝜇)ℂ. It is known as the

trigonometric system. Therefore, for 𝑓 ∈ 𝐿2 −𝜋, 𝜋 we have

𝑓 𝑡 = 𝑓 (𝑛)𝑛∈ℤ 𝑒𝑖𝑛𝑡 (Fourier expansion)

convergence in ∙ 2, where

𝑓 (𝑛):= 1

2𝜋∫ 𝑓(𝑡)𝑒−𝑖𝑛𝑡𝑑𝑡

𝜋

−𝜋

Similarly, the family 1

2, cos 𝑛𝑡 , sin(𝑛𝑡) : 𝑛 ∈ ℕ is a Hilbert basis of

𝐿2( −𝜋, 𝜋 , , 𝜇)ℝ. Therefore, for 𝑓 ∈ 𝐿2 −𝜋, 𝜋 we have

𝑓(𝑡) =𝑎0

2+ (𝑎𝑛cos 𝑛𝑡 + 𝑏𝑛 sen 𝑛𝑡 )∞

𝑛=1 (Fourier expansion)

Convergence in ∙ 2, where:

𝑎0 =1

𝜋∫ 𝑓(𝑡) 𝑑𝑡

𝜋

−𝜋 ; 𝑎𝑛 =

1

𝜋∫ 𝑓(𝑡) cos 𝑛𝑡 𝑑𝑡

𝜋

−𝜋; 𝑏𝑛=

1

𝜋∫ 𝑓(𝑡) sen 𝑛𝑡 𝑑𝑡

𝜋

−𝜋

Hilbert spaces : The prototype

Fact: If 𝐻 is a Hilbert space and 𝐵 ≔ *𝑒𝑖: 𝑖 ∈ 𝑆+ is a Hilbert basis then, every

𝑢 ∈ 𝐻 has a unique expansion given by

𝑢 = 𝑢, 𝑒𝑖 𝑒𝑖𝑖∈𝐼 and

𝑢 = 𝑢, 𝑒𝑖2

𝑖∈𝐼 .

Consequently, the mapping 𝒖 → 𝒖, 𝒆𝒊 𝒊∈𝑰 defines an isometric isomorphism

from 𝑯 onto 𝒍𝟐(𝑰). Therefore, 𝑙2(𝐼) becomes the Hilbert space prototype.

Example: The mapping 𝐿2 −𝜋, 𝜋 ℂ → 𝑙2(ℤ), given by 𝑓 → 𝑓 (𝑛)n∈ℤ

defines an

isometric isomorphism.

For this reason the Hilbert spaces have “coordinates”, as well as the euclidean

space (which is also a Hilbert space) has its own coordinates.

Theorem: Two Hilbert spaces are isometrically isomorphic, if and only if, they have

the same Hilbert dimension.

(that is that an orthonormal basis of one of this spaces has the same cardinality

of an orthonormal basis of the other one).

Hilbert spaces: the Riesz-Fréchet theorem

For general Banach spaces, this was open problem along

more of 20 years. The answer was the Hahn-Banach

theorem (a cornerstone theorem of the Functional Analysis). Fréchet, Maurice

1878-1973

If dim 𝐻 = 𝑛 then 𝐻 = K𝑛. Fix an orthonormal basis 𝐵 = *𝑒1,…, 𝑒𝑛+. If 𝑓: 𝐻 → K

is linear then, for 𝑢 = 𝛼𝑖𝑒𝑖𝑛𝑖=1 , we have

𝑓 𝑢 = 𝛼𝑖𝑓(𝑒𝑖)𝑛𝑖=1 = 𝑢, 𝑣

where 𝑣 = 𝑓(𝑒𝑖)𝑒𝑖𝑛𝑖=1 . Therefore 𝑓 = ∙, 𝑣 .

Conversely, 𝑓 = ∙, 𝑣 is a continuous linear functional, obviously. Thus,

𝐻∗ = ∙, 𝑣 : 𝑣 ∈ 𝐻 .

If 𝑑𝑖𝑚𝐻 = ∞ then, ∙, 𝑣 : 𝑣 ∈ 𝐻 ⊆ 𝐻∗ trivially. What F. Riesz and M. Fréchet

showed (in independent papers published in Comptes Redus in 1907) is that, in

fact, ∙, 𝑣 : 𝑣 ∈ 𝐻 = 𝐻∗.

Note also that if 𝑓𝑣: = ∙, 𝑣 , then 𝑓𝑣 = 𝑣 .

Let 𝐻 be a Hilbert space over K. Who is 𝐻∗? This is to determine the set of continuous linear functionals 𝑓: 𝐻 → K.

Espacios de Hilbert: El teorema de Riesz-Fréchet

Riesz-Fréchet theorem (1907): Let 𝐻 be a Hilbert space and let 𝑓 ∈ 𝐻∗. Then, there

exists a unique 𝑣 ∈ 𝐻 such that 𝑓 = 𝑓𝑣 = ∙, 𝑣 . Moreover, 𝑓 = 𝑣 .

Proof: Let 𝑓 ∈ 𝐻∗. If 𝑓 = 0, then obviously 𝑓 = 𝑓0 = ∙, 0 . If 𝑓 ≠ 0, then ker𝑓 is a

proper closed subspace of 𝐻, so that ker𝑓 ⊥ is nonzero. Therefore, by the OPT

there exists 𝜔 ∈ ker𝑓 ⊥ such that 𝑓(𝜔) ≠ 0. Now it is easy to check that 𝑣 =𝑓(𝜔)

𝜔 2 𝜔

is the vector that we look for, and the result follows.

Corollary: If 𝐻 is a Hilbert space, then there exists an isometric conjugate-linear

bijection between 𝐻 and 𝐻∗.

Proof: Consider the map 𝐻∗→ 𝐻 given by 𝑓𝑣 = ∙, 𝑣 → 𝑣.

Corollary: If 𝐻 is a Hilbert space, then 𝐻∗ is also a Hilbert space.

Proof: Define 𝑓𝑢, 𝑓𝑣 ≔ 𝑢, 𝑣 and check it.

Corollary: Every Hilbert space is reflexive (i. e. 𝐻 = 𝐻∗∗).

Corollary: Any inner product space may be completed to a Hilbert space.

Hilbert spaces: The weak topology

Corollary: If in a Hilbert space, 𝑢𝑛

∙𝑢 ⇒ 𝑢𝑛

𝜔→ 𝑢.

The converse does not hold. For instance, in 𝑙2 we have that 𝑒𝑛, 𝑣 →0 = 0, 𝑣 ,

so that 𝑒𝑛

𝜔→ 0, meanwhile 𝑒𝑛 − 𝑒𝑚 = 2.

Corollary: The (norm)-closed unit ball in a Hilbert space is weakly compact.

Corollary: If 𝐻 is a Hilbert space and 𝑀 is a subspace, then every bounded linear

functional 𝑓: 𝑀 →K admits a bounded linear extension 𝑓 : 𝐻 →K with 𝑓 = 𝑓 .

Definition: A sequence 𝑢𝑛, in a Hilbert 𝐻, converges weakly to 𝑢 ∈ 𝐻 (and we

write 𝑢𝑛

𝜔→ 𝑢 ) whenever 𝑢𝑛, 𝑣 → 𝑢, 𝑣 , for every 𝑣 ∈ 𝐻.

Bounded linear functionals in a Hilbert space do exist in abundance, as we have

shown. Consider the smallest topology that makes that all of them are continuous.

This is the so-called weak topology (the strong topology is the norm topology).

Unless that dim 𝐻 < ∞ , the weak topology is not normable.

From the continuity of the inner product we deduce the following result.

Operators in Hilbert spaces: The adjoint operator Let 𝐻1 and 𝐻2 be Hilbert spaces and let 𝑇 ∈ 𝐿 𝐻1, 𝐻2 . As a consequence of the

Riesz-Fréchet theorem, there exists a unique bounded linear operator 𝑇∗: 𝐻2 → 𝐻1

such that

𝑇𝑢, 𝜔 = 𝑢, 𝑇∗ 𝜔 , (𝑢 ∈ 𝐻1, 𝜔 ∈ 𝐻2).

Note that

𝑇∗ 𝜔 2 = 𝑇∗ 𝜔 , 𝑇∗ 𝜔 = 𝑇𝑇∗ 𝜔 , 𝜔 ≤ 𝑇 𝑇∗ 𝜔 𝜔 ,

Therefore

𝑇∗ 𝜔 ≤ 𝑇 𝜔 .

Consequently, 𝑇∗ is continuous and 𝑇∗ ≤ 𝑇 .

Definition: Let 𝐻1 and 𝐻2 be Hilbert spaces and let 𝑇: 𝐻1 → 𝐻2 be a bounded linear

operator. The adjoint operador of 𝑇 is defined as the unique bounded linear

operator 𝑇∗: 𝐻2 → 𝐻1 such that 𝑇𝑢, 𝜔 = 𝑢, 𝑇∗ 𝜔 , (𝑢 ∈ 𝐻1, 𝜔 ∈ 𝐻2).

Fact: 𝑇∗∗ = 𝑇. Indeed, if 𝑢 ∈ 𝐻2, and 𝜔 ∈ 𝐻1, then:

𝑇∗𝑢, 𝜔 = 𝜔, 𝑇∗𝑢 = 𝑇𝜔, 𝑢 = 𝑢, 𝑇𝜔 = 𝑢, 𝑇∗∗ 𝜔 . Since 𝑇∗∗ 𝜔 is unique, we obtain that 𝑇∗∗ 𝜔 = 𝑇𝜔.

Operators in Hilbert spaces: The adjoint operator

Proposition: Let 𝐻1 and 𝐻2 be Hilbert spaces and let 𝑇, 𝑆 ∈ 𝐿(𝐻1, 𝐻2). Then:

i) 𝑇∗∗ = 𝑇 ii) 𝛼𝑇 ∗ = 𝛼 𝑇∗ (𝛼 ∈ K)

iii) 𝑇 + 𝑆 ∗ = 𝑇∗ + 𝑆∗

Moreover, if 𝐻3 is a Hilbert space and 𝑅 ∈ 𝐿(𝐻2, 𝐻3), then (𝑅𝑇)∗= 𝑇∗𝑅∗.

Since 𝑇∗∗ = 𝑇 and 𝑇∗∗ ≤ 𝑇∗ , we obtain that 𝑇∗ = 𝑇 .

Indeed:

Proposition: Let 𝐻1 and 𝐻2 be Hilbert spaces and let 𝑇 ∈ 𝐿(𝐻1, 𝐻2). Then,

𝑇∗ = 𝑇 = 𝑇∗𝑇 = 𝑇𝑇∗ (Gelfand-Naimark).

Fact: The completeness is essential for the existence of the adjoint.

Example: Let 𝑇: 𝑐00 → 𝑐00 be the operador 𝑇𝑥 = 𝑥𝑛

𝑛∞𝑛=1 𝑒1. That is

𝑥 = 𝑥1, 𝑥2 … . . 𝑥𝑛, … → 𝑇𝑥 = * 𝑥𝑛

𝑛∞𝑛=1 , 0 … , 0, … +.

Let 𝑦 = *𝑦𝑛+ with 𝑦1 ≠ 0. Let 𝑇∗𝑦 = 𝑧𝑛 . Then 𝑇∗𝑦 ∉ 𝑐00.

Operators in Hilbert spaces: The adjoint operator

From the above result we obtain straightforwardly the following proposition.

Examples: Let 𝐻 be a Hilbert space and let B = *𝑒𝑖+𝑖∈𝐼 be a Hilbert basis.

i) If 𝑇 ∈ 𝐿 𝐻 is diagonal (i.e. 𝑇𝑒𝑖 = 𝜆𝑖𝑒𝑖), then 𝑇∗𝑒𝑖 = 𝜆𝑖 𝑒𝑖 .

ii) If 𝑇 ∈ 𝐿 𝐻 is the orthogonal projection on a closed subspace, then 𝑇 = 𝑇∗.

Notation: 𝐿(𝐻) = 𝐿(𝐻, 𝐻).

Examples:

i) If, repect to the basis B, the matrix associated to 𝑇 ∈ 𝐿 𝑙2 is 𝑎𝑖𝑗 , then the

matrix associated to T∗ is 𝑏𝑖𝑗 = 𝑎𝑖𝑗 .

ii) If 𝑇 ∈ 𝐿 𝐿2 𝑎, 𝑏 is an integral operator with kernel 𝑘 ∈ 𝐿2( 𝑎, 𝑏 × 𝑎, 𝑏 ), then

𝑇∗ is an integral operator with kernel 𝑘∗ 𝑡, 𝑠 = 𝑘 𝑠, 𝑡 .

Proposition: Let 𝐻1 and 𝐻2 be Hilbert spaces and let 𝑇 ∈ 𝐿(𝐻1, 𝐻2). Then,

i) 𝑇∗ is injective ⟺ 𝑇 has dense range. (Ker 𝑇∗ = (Im 𝑇)⊥) ii) 𝑇 is injective ⟺ 𝑇∗has dense range. (Ker 𝑇 = (Im𝑇∗)⊥) iii) 𝑇 is biyective ⟺ T∗ is bijective, in whose case (T∗)−1= (T−1)∗.

Operators in Hilbert spaces: The adjoint operator

Proposición: Let S, 𝑇 ∈ 𝐿 𝐻 be selft-adjoint operators. Then, i) 𝑆 + 𝑇 is selft-adjoint.

ii) 𝑆𝑇 selft-adjoint ⟺ 𝑆𝑇 = 𝑇𝑆.

Fact: From now on, we will consider operators from a Hilbert space into itself, that is

the space 𝐿 𝐻 : = 𝐿 𝐻, 𝐻 of bounded linear operators 𝑇: 𝐻 → 𝐻.

Definition: 𝑇 ∈ 𝐿(𝐻) is a self-adjoint operator if 𝑇 = 𝑇∗.

Recall that if 𝑇 ∈ 𝐿 𝐻 , then 𝑇∗ is the unique operator 𝐿(𝐻) such that 𝑇𝑢, 𝜔 = 𝑢, 𝑇∗ 𝜔 , (𝑢, 𝑣 ∈ 𝐻).

Proposition: Let 𝑇 ∈ 𝐿 𝐻 . If 𝑇 = 𝑇∗, then 𝐻 = ker 𝑇 ⨁ 𝐼𝑚𝑇.

Proposition: Let 𝐻 be a complex Hilbert space and let 𝑇 ∈ 𝐿 𝐻 . Then:

𝑇 = 𝑇∗ ⟺ 𝑇𝑢, 𝑢 ∈ ℝ, for every 𝑢 ∈ 𝐻. Moreover, if 𝑇 is selft-adjoint, then

𝑇 = sup 𝑇𝑢, 𝑢 : 𝑢 ≤ 1 = sup*|⟨Tu, u⟩|: ‖u‖ = 1+.

Operators in Hilbert spaces: The adjoint operator

Proposition: If 𝑇 ∈ 𝐿(𝐻) is such that 𝑇2 = 𝑇 (i.e. 𝑇 is a projection), then,

T normal ⟺ T is a orthogonal projection ⟺ 𝑇 = 1.

Proposition: Let 𝑇 ∈ 𝐿 H . Then 𝑇 normal ⟺ 𝑇𝑢 = 𝑇∗𝑢 𝑢 ∈ 𝐻 .

Proposition: Let 𝐻 be a complex Hilbert space and let 𝑇 ∈ 𝐿(𝐻). Then, there exist

self-adjoint operators R, S ∈ 𝐿 H (which are unique) such that 𝑇 = 𝑅 + 𝑖𝑆.

Proof: 𝑇 =𝑇+𝑇∗

2+ 𝑖

𝑇−𝑇∗

2𝑖.

The unique self-adjoint operators R, S ∈ 𝐿 𝐻 , such that 𝑇 = 𝑅 + 𝑖𝑆, are called

the real part and the imaginary part of 𝑇, respectivelty.

Definition: 𝑇 ∈ 𝐿 H is called normal if 𝑇𝑇∗ = 𝑇∗ 𝑇.

Proposition: Let 𝐻 be a complex Hilbert space. Then 𝑇 ∈ 𝐿 𝐻 is normal if, and

only if, its real and imaginary parts commute.

𝑇 ∈ 𝐿 𝐻 self-adjoint ⟹ 𝑇 normal

𝑇 ∈ 𝐿(𝐻) diagonal ⟹ 𝑇 normal

Operators in Hilbert spaces: The spectral equation

Goal: To solve the spectral equation 𝑇 − 𝜆𝐼 𝑥 = 𝑦.

Let 𝑇: ℂ𝑛 → ℂ𝑛 be a linear operator. If 𝐴 is the matrix associated to 𝑇 respect to the

canonical basis, then we like to solve the equation

𝐴 − 𝜆𝐼 𝑥 = 𝑦, where 𝐴 is an 𝑛 × 𝑛 matrix, 𝐼 is the identity matrix, and 𝑥 and 𝑦 are the

coordinates of the corresponding vectors (where 𝑦 is known and 𝑥 is not).

If det 𝐴 − 𝜆𝐼 ≠ 0, then 𝐴 − 𝜆𝐼 is an invertible matrix and the given equation has

a unique solution given by 𝑥 = 𝐴 − 𝜆𝐼 −1𝑦. If det 𝐴 − 𝜆𝐼 = 0, then 𝑇 − 𝜆𝐼 is not injective, nor surjective. Therefore the

system has a solution (not unique) if, and only if, 𝑦 ∈ 𝐼𝑚 𝑇 − 𝜆𝐼 .

Thus, determining the set of all 𝜆 ∈ ℂ such that 𝑇 − 𝜆𝐼 is not invertible is relevant:

*𝜆 ∈ ℂ ∶ det 𝐴 − 𝜆𝐼 = 0+.

Definition: 𝑇 ∈ 𝐿 𝐻 is invertible if there exists 𝑅 ∈ 𝐿 𝐻 such that 𝑇𝑅 = 𝑅𝑇 = 𝐼. If turns out that 𝑅 is unique (whenever it exists). We denote 𝑅 = 𝑇−1.

Operators in Hilbert spaces: The spectrum

If 𝑇: ℂ𝑛 → ℂ𝑛 is a linear operator, and if 𝐴 is its matrix respecto to the canonical

basis, then

𝜎 𝑇 = *𝜆 ∈ ℂ ∶ 𝑇 − 𝜆𝐼 is not invertible+ = 𝜎𝑝 𝑇 = *𝜆 ∈ ℂ ∶ det 𝐴 − 𝜆𝐼 = 0+.

In fact:

𝑇 invertible ⇔ 𝑇 injective ⇔ 𝑇 surjective

Thus,

𝜎𝑝 𝑇 = *𝜆 ∈ ℂ ∶ det 𝐴 − 𝜆𝐼 = 0+ = *𝜆 ∈ ℂ ∶ 𝑇 − 𝜆𝐼 is not injective+.

Definition: Let 𝐻 be a complex Hilbert space. The spectrum of 𝑇 ∈ 𝐿 H is the set

𝜎 𝑇 = *𝜆 ∈ ℂ ∶ 𝑇 − 𝜆𝐼 is not invertible+

Definition: Let 𝐻 be a complex Hilbert space. The pointwise spectrum of 𝑇 ∈ 𝐿(𝐻) is the set given by

𝜎𝑝 𝑇 = *𝜆 ∈ ℂ ∶ 𝑇 − 𝜆𝐼 is not injective+.

The elements of 𝜎𝑝 𝑇 are called eigenvalues of 𝑇.

If 𝑥 ∈ ker 𝑇 − 𝜆𝐼 then 𝑥 is an eigenvector of 𝑇 associated to the eigenvalue 𝜆.

Fact: Let 𝐻 be a finite-dimensional complex Hilbert space. Then, 𝜎 𝑇 = 𝜎𝑝 𝑇 , for

every 𝑇 ∈ 𝐿 𝐻 (i. e. the spectrum and the pointwise spectrum coincide).

Operators in Hilbert spaces: The spectrum

Fact: If 𝐻 is an infinite-dimensional complex Hilbert space, then 𝜎𝑝 𝑇 ⊆ 𝜎 𝑇

and these sets do not need to coincide.

Fact: Let 𝑇 ∈ 𝐿 𝐻 . By the Banach isomorphism theorem we have that there exists

𝑇−1 ∈ 𝐿(𝐻) such that 𝑇𝑇−1 = 𝑇−1𝑇 = 𝐼 if, and only if, 𝑇 is bijective.

Definition: Let 𝐻 be a complex Hilbert space. The surjective spectrum of 𝑇 ∈ 𝐿 H

is defined by 𝜎𝑠𝑢 𝑇 = *𝜆 ∈ ℂ ∶ 𝑇 − 𝜆𝐼 is not surjective+.

Proposition: Let 𝐻 be a complex Hilbert space. For every 𝑇 ∈ 𝐿 H we have

𝜎 𝑇 = 𝜎𝑝 𝑇 ∪ 𝜎𝑠𝑢 𝑇 .

Example: If 𝜋: 𝐻 → 𝑀 is the projection of 𝐻 over a closed subspace 𝑀, then

𝜎𝑝 𝑇 = 0,1 . Moreover 𝑀 is the invariant subspace associated to the eigenvalue

0 and 𝑀⊥ the one associated to the eigenvalue 1.

Example: The Volterra operator 𝑇: 𝐿2,0,1- → 𝐿2 0,1 is such that 𝜎𝑝 𝑇 = ∅.

Operators in Hilbert spaces: The spectrum

Remark: If 𝐻 is a real Hilbert space, then the set *𝜆 ∈ ℝ ∶ 𝑇 − 𝜆𝐼 is not bijective+ may

be empty (in whose case, no information is provided).

Theorem: Let 𝐻 be a complex Hilbert space, and let 𝑇 ∈ 𝐿 H . Then, 𝜎 𝑇 is a

non-empty compact subset of ℂ.

Definition: Let 𝐻 be a real Hilbert space. The complexification of 𝐻 is defined as the

complex Hilbert space 𝐻ℂ ≔ 𝐻⨁𝑖𝐻. Moreover, the spectrum of 𝑇 ∈ 𝐿(𝐻) is defined

as the spectrum of the operator 𝑇ℂ ∈ 𝐿(𝐻ℂ) given by 𝑇ℂ 𝑢 + 𝑖𝑣 = 𝑇 𝑢 + 𝑖𝑇 𝑣 .

Example: 𝑇 ∈ 𝐿 ℂℝ given by 𝑇 𝑥 = 𝑖𝑥.

In fact, (𝑇 − 𝜆𝐼)𝑥 = 𝑖 − 𝜆 𝑥 always is bijective because if 𝜆 ∈ ℝ, then 𝑖 − 𝜆 ≠ 0.

Agreement: From now on, only complex Hilbert spaces will be considered.

Theorem: Let 𝐻 be a Hilbert space (either real or complex). If 𝑇 ∈ 𝐿(𝐻) then 𝜎 𝑇 is

a non-empty compact subset of ℂ.

Operators in Hilbert spaces: The spectrum

Proposition : If 𝑇 ∈ 𝐿 H then 𝜎 𝑇∗ = *λ ∶ 𝜆 ∈ 𝜎 𝑇 + (the conjugate set of 𝜎 𝑇 ).

Corollary: If 𝑇 ∈ 𝐿 H is normal, then 𝑇 = max 𝜆 : 𝜆 ∈ 𝜎 𝑇 .

Proposition: If 𝑇 ∈ 𝐿 H is normal then,

𝜆 ∈ 𝜎(𝑇) ⟺ there exists 𝑥𝑛 with 𝑥𝑛 = 1 such that (𝑇 − 𝜆𝐼)𝑥𝑛 → 0.

Corollary: If 𝑇 ∈ 𝐿 H is self-adjoint then 𝜎 𝑇 ⊆ ℝ.

Proof: 𝑇 − 𝜆𝐼 invertible ⇔ 𝑇 − 𝜆𝐼 𝑅 = 𝑅 𝑇 − 𝜆𝐼 = 𝐼 ⇔ 𝑅∗ 𝑇 − 𝜆𝐼 ∗ = 𝑇 − 𝜆𝐼 ∗𝑅∗ = 𝐼∗

⇔ 𝑅∗ 𝑇∗ − 𝜆 𝐼 = 𝑅∗ 𝑇∗ − 𝜆 𝐼 = 𝐼 ⇔ 𝑇∗ − 𝜆 𝐼 invertible.

Note: This is equivalent to the following fact:

𝜆 ∈ ℂ\𝜎(𝑇) ⟺ there exists 𝑐 > 0 such that 𝑇 − 𝜆𝐼 𝑥 ≥ 𝑐 𝑥 , 𝑥 ∈ 𝐻 .

Proof: This is a consequence of the fact that if 𝑇 ∈ 𝐿 H is normal, then

𝑇 = 𝑠𝑢𝑝 𝑥 =1 𝑇𝑥, 𝑥 (the numerical radius).

Compact operators on Hilbert spaces Let (Ω, Σ, 𝜇) be a measure space and let 𝑘(𝑠, 𝑡) ∈ 𝐿2(Ω × Ω, Σ × Σ, 𝜇 × 𝜇). Then, the

integral operator with kernel 𝑘(𝑠, 𝑡) is the operator 𝑇: 𝐿2(𝜇) → 𝐿2(𝜇) given by

𝑇𝑥 𝑠 = ∫ 𝑘 𝑠, 𝑡 𝑥 𝑡 𝑑𝜇(𝑡)Ω

(𝑠 ∈ Ω, 𝑥 ∈ 𝐿2(𝜇))

In Quantum Mechanics the integral equation 𝑇 − 𝜆𝐼 𝑥 𝑠 = 𝑦 𝑠 , where 𝑇 is the

integral operator with kernel 𝑘, is essential.

Proposition: Let 𝑋 and 𝑌 be normed spaces and let 𝑇 ∈ 𝐿 X, Y . Then, the

following assertions are equivalent:

i) 𝑇 is compact,

ii) For every sequence 𝑥𝑛 in 𝑋 with 𝑥𝑛 = 1, the sequence 𝑇𝑥𝑛 has a convergent

subsequence in 𝑌, iii) 𝑇(𝐵𝑋) is compact in 𝑌 (where 𝐵𝑋 denotes the closed unit ball of 𝑋).

Definition: Let 𝑋 and 𝑌 be normed spaces. An operator 𝑇 ∈ 𝐿 X, Y is compact if,

for every bounded sequence 𝑥𝑛 in 𝑋, the sequence 𝑇𝑥𝑛 has a convergent

subsequence in 𝑌.

Example: The integral operator 𝑇: 𝐿2(𝜇) → 𝐿2(𝜇) with kernel 𝑘 is compact.

Compact operators on Hilbert spaces

Proposition: Let 𝑋 and 𝑌 be normed spaces and let 𝑇 ∈ 𝐾 X, Y . Then,

i) 𝑇(𝑋) is a separable subspace of 𝑌. ii) If 𝑌 is a Hilbert space, and if 𝐵 = *𝑒𝑛: 𝑛 ∈ ℕ+ is a Hilbert basis, then 𝑇 = 𝑙𝑖𝑚𝜋𝑛𝑇 where 𝜋𝑛 is the orthogonal projection on 𝐿𝑖𝑛 𝑒1, … , 𝑒𝑛 .

Notation: Let 𝑋 and 𝑌 be normed spaces. Let denote 𝐾(𝑋, 𝑌)= *𝑇 ∈ 𝐿 𝑋, 𝑌 : 𝑇 is compact+.

Example: If 𝑇 ∈ 𝐿 X, Y is a finite rank opertor, then 𝑇 is compact.

If 𝑥𝑛 = 1, then 𝑇𝑥𝑛 is a bounded sequence in a finite dimensional normed space

𝑌, and hence it has a convergent subsequence.

Example: If 𝐻 is an infinite dimensional Hilbert space, then the identity operator

𝐼: 𝐻 → 𝐻 is not compact.

In fact, if 𝑒𝑛 is an orthonormal sequence, then 𝐼𝑒𝑛 does not have a convergent

subsequence.

Proposition: 𝐾 X, Y is closed whenever 𝑌 is a Banach space.

Compact operators on Hilbert spaces

Corollary (Jordan): Let 𝐻 and 𝐾 be Hilbert spaces. Then, 𝐾 𝐻, 𝐾 = 𝐹(𝐻, 𝐾).

Notation: Let 𝐻 and 𝐾 be Hilbert spaces. Let denote

𝐹 𝐻, 𝐾 = *𝑇 ∈ 𝐿 𝐻, 𝐾 : 𝑇 has finite rank+.

Note that the integral operator 𝑇 with kernel 𝑘 is the limit of the sequence of finite-

rank operators 𝑇𝑛, where 𝑇𝑛 is the integral operators with kernel 𝑘𝑛 for a sequence

𝑘𝑛 of simple functions with 𝑘𝑛 → 𝑘.

Corollary (Theorem of the adjoint): Let 𝐻 and 𝐾 Hilbert spaces. Then,

𝑇 ∈ 𝐾 𝐻, 𝐾 ⟺ 𝑇∗ ∈ 𝐾 𝐻, 𝐾 .

Theorem: Let 𝑇 ∈ 𝐿(𝐻) be a compact operator and let 𝜆 ≠ 0. Then,

i) dim ker 𝑇 − 𝜆 𝐼 < ∞ (Theorem of the kernel) (Riesz)

ii) (𝑇 − 𝜆𝐼)(𝐻) is closed (Theorem of the rank).

Corollary: Let 𝑇 ∈ 𝐿(𝐻) a compact operators and let 𝜆 ≠ 0. Then,

(𝑇 − 𝜆𝐼)(𝐻) = 𝐻 ⟺ ker 𝑇 − 𝜆 𝐼 = *0+. Consequently,

𝜎𝑠𝑢(𝑇)\*0+ = 𝜎𝑝(𝑇)\*0+ = 𝜎(𝑇)\*0+.

In finite dimension 𝜎𝑠𝑢(𝑇) = 𝜎𝑝(𝑇) = 𝜎(𝑇).

Compact operators on Hilbert spaces

Theorem (Fredholm alternative): Let 𝑇 ∈ 𝐿(𝐻) be a compact operator and let

𝜆 ≠ 0. Consider the following equations:

(a) 𝑇 − 𝜆 𝐼 𝑥 = 𝑦 (b) 𝑇∗ − 𝜆 𝐼 𝑥 = 𝑦

(c) 𝑇 − 𝜆 𝐼 𝑥 = 0 (d) 𝑇∗ − 𝜆 𝐼 𝑥 = 0

Then either

i) the equations (a) and (b) has a solution 𝑥 and 𝑥 , for every 𝑦, 𝑦 ∈ 𝐻, resp.

ii) or the homogeneous system of equations (c) and (d) have a non-trivial

solution.

In the case (i), the solutions 𝑥 and 𝑥 are unique and depend continously on 𝑦

and 𝑦 respectively.

In the case (ii), the equation (a) has a unique solution 𝑥 if, and only if, 𝑦 is

orthogonal to all the solutions of (d). Similarly (b) has a unique solution 𝑥 if, and only if, 𝑦 is orthogonal to all the solutions of (a).

Proof: If 𝜆 ∉ 𝜎(𝑇) then we have (i) obviously, because 𝜎(𝑇) = 𝜎(𝑇∗). In fact,

𝑥 = (𝑇 − 𝜆𝐼)−1𝑦 meanwhile 𝑥 = (𝑇 − 𝜆𝐼)−1𝑦 .

Otherwise, 𝜆 ∈ 𝜎(𝑇) ∖ *0+ = 𝜎𝑝(𝑇) ∖ *0+, and hence (c) and (d) have a non-trivial

solution. Moreover, (a) has a solution ⟺ 𝑦 ∈ 𝑇 − 𝜆 𝐼 𝐻 = ker 𝑇∗ − 𝜆 𝐼⊥.

Similarly (b) has a solution ⟺ 𝑦 ∈ 𝑇∗ − 𝜆 𝐼 𝐻 = ker 𝑇 − 𝜆 𝐼 ⊥.

Compact operators on Hilbert spaces

Corollary: If 𝑇 ∈ 𝐿(𝐻) is a self-adjoint compact operator, then 𝑇 or − 𝑇 is an

eigenvalue of 𝑇.

Indeed it is known that if 𝑇 is normal, then 𝑇 = max 𝜆 : 𝜆 ∈ 𝜎 𝑇 .

Remarks:

i) Recall that if 𝑇 is compact, then 𝜎𝑝(𝑇)\*0+ = 𝜎(𝑇)\*0+.

ii) The restriction of a self-adjoint compact operator to an invariant subspace is a

self-adjoint compact operator.

Diagonalization of a selft-adjoint compact operator

Let 𝑇 ∈ 𝐿 𝐻 a self-adjoint compact operator (𝑇 ≠ 0).

Let 𝜆1 ∈ 𝜎𝑝(𝑇) such that 𝜆1 = 𝑇 . Let 𝑢1 with 𝑢1 = 1 such that 𝑇 − 𝜆1𝐼 𝑢1 = 0.

Let 𝐻1 = 𝐻 and 𝐻2 = 𝑢 ∈ 𝐻1: 𝑢 ⊥ 𝑢1 = 𝑢1⊥. If 𝑢 ∈ 𝐻2, then 𝑢, 𝑢1 = 0 and hence,

0 = 𝜆1 𝑢, 𝑢1 = 𝑢, 𝜆1𝑢1 = 𝑢, 𝑇𝑢1 = 𝑇∗𝑢, 𝑢1 = 𝜆1 𝑢, 𝑢1 = 𝑇𝑢, 𝑢1 ,

Thus, 𝑇(𝐻2)= 𝐻2, so that 𝑇∕𝐻2 is self-adjoint and compact.

Compact operators on Hilbert spaces

Repeat the process with 𝑇∕𝐻2.

Let 𝜆2 ∈ 𝜎𝑝(𝑇∕𝐻2) ⊆ 𝜎𝑝(𝑇) such that 𝜆2 = 𝑇∕𝐻2

.

Note that 𝜆2 ≤ 𝜆1 because 𝑇∕𝐻2 ≤ 𝑇 .

Let 𝐻3 = 𝑢 ∈ 𝐻2: 𝑢 ⊥ 𝑢2 . Then 𝑇∕𝐻2 is self-adjoint and compact and 𝑇∕𝐻2

(𝐻3) = 𝐻3.

Note that

𝐻2 = 𝑢 ∈ 𝐻1: 𝑢 ⊥ 𝑢1 = 𝑢1⊥

𝐻3 = 𝑢 ∈ 𝐻2: 𝑢 ⊥ 𝑢2 = 𝑢1⊥∩ 𝑢2

⊥ = 𝑢1, 𝑢2⊥

Reiterating the process we obtain nonzero eigenvalues 𝜆1, 𝜆2,…, 𝜆𝑛 such that

𝜆𝑛 ≤ ⋯ ≤ 𝜆2 ≤ 𝜆1 ,

with unital eigenvectors 𝑢1, 𝑢2,…, 𝑢𝑛, and closed invariant subspaces 𝐻1, 𝐻2,…, 𝐻𝑛,

where 𝐻𝑗+1 = 𝑢 ∈ 𝐻𝑗: 𝑢 ⊥ 𝑢𝑗 is such that 𝐻𝑛 ⊆ ⋯ ⊆ 𝐻2 ⊆ 𝐻1 and 𝜆𝑗 = 𝑇∕𝐻𝑗.

The process stops only when 𝑇∕𝐻𝑛+1= 0.

Since 𝐻𝑛+1 = 𝑢1, 𝑢2,…, 𝑢𝑛⊥ and 𝐻 = 𝑢1, 𝑢2,…, 𝑢𝑛 ⨁ 𝑢1, 𝑢2,…, 𝑢𝑛

⊥ (OPT) we

have: if 𝑢 ∈ 𝐻, then 𝑢 = 𝑢, 𝑢𝑖 𝑢𝑖 + 𝑣𝑛𝑖=1 with 𝑣 ∈ 𝐻𝑛+1. Thus, if 𝑇∕𝐻𝑛+1

= 0 then

𝑇𝑢 = 𝜆𝑖 𝑢, 𝑢𝑖 𝑢𝑖𝑛𝑖=1

(this is the case, particularly, if 𝑑𝑖𝑚𝐻<∞).

Compact operators on Hilbert spaces

If the process does not stop, then we obtain a sequence of eigenvalues 𝜆𝑛→ 0.

Indeed, if 𝜆𝑛→ 𝜆 > 0 then 1

𝜆𝑛 𝑢𝑛 is bounded so that 𝑇(

1

𝜆𝑛𝑢𝑛)= 𝑢𝑛 has a

covergent subsequence which contradicts that 𝑢𝑛 − 𝑢𝑚 = 2.

Hence, in this case, for every 𝑛 ∈ ℕ we have that

𝐻 = 𝑢1, 𝑢2,…, 𝑢𝑛 ⨁ 𝑢1, 𝑢2,…, 𝑢𝑛⊥.

If 𝑢 = 𝑢, 𝑢𝑖 𝑢𝑖 + 𝑣𝑛𝑛𝑖=1 with 𝑣𝑛 ∈ 𝑢1, 𝑢2,…, 𝑢𝑛

⊥ = 𝐻𝑛+1, then we obtain that

𝑇𝑣𝑛 = 𝑇∕𝐻𝑛+1𝑣𝑛 ≤ 𝑇∕𝐻𝑛+1

𝑣𝑛 = 𝜆𝑛+1 𝑣𝑛 ≤ 𝜆𝑛+1 𝑢 → 0,

And hence,

𝑇𝑢 = 𝜆𝑖 𝑢, 𝑢𝑖 𝑢𝑖

∞

𝑖=1

Note that if 𝜆 is a nonzero eigenvalue of 𝑇, then 𝜆 ∈ 𝜆𝑛: 𝑛 ∈ ℕ . Otherwise, if 𝑢

is an associated eigenvalue then, 𝜆𝑢 = 𝑇𝑢 = 0, which is impossible.

Compact operators on Hilbert spaces: spectral theorem

Theorem (spectral theorem for compact self-adjoint operators):

Let 𝑇 ∈ 𝐿(𝐻) be a compact self-adjoint operator. Then 𝑇 is diagonalizable.

Indeed, one of the following assertions holds:

i) There exist eignevalues 𝜆1, 𝜆2,…, 𝜆𝑛 and a system of associated

orthonormal eigenvectors 𝑢1, 𝑢2,…, 𝑢𝑛 such that, for every 𝑢 ∈ 𝐻,

𝑇𝑢 = 𝜆𝑖 𝑢, 𝑢𝑖 𝑢𝑖

𝑛

𝑖=1

(uniform convergence over the compact subsets of 𝐻).

ii) There exists a sequence 𝜆𝑛 of eigenvalues such that 𝜆𝑛→ 0, and a

sequence of associated orthonormal eigenvectors 𝑢𝑛 such that, ∀𝑢 ∈ 𝐻,

𝑇𝑢 = 𝜆𝑖 𝑢, 𝑢𝑖 𝑢𝑖

∞

𝑖=1

(uniform convergence over the compact subsets of 𝐻).

In (i) as well as in (ii), if 𝜆 is a non-zero eigenvalue, then 𝜆 ∈ 𝜆1, 𝜆2 … . Moreover, the dimension of the invariant subspace associated to 𝜆

coincides with the number of times that 𝜆 appears in 𝜆1, 𝜆2 … .

Compact operators on Hilbert spaces: spectral theorem

Corollary: 𝑇 ∈ 𝐿 𝐻 is a compact self-adjoint operator ⟺ 𝑇 is diagonalizable, i.e.

𝑇 = 𝜆𝑖 ∙, 𝑢𝑖 𝑢𝑖

𝑖

for a countable family of real numbers 𝜆1, 𝜆2 … and a orthonormal system

𝑢1, 𝑢2 … .

Rearranging the above sum, fix 𝑘 and denote 𝑃𝜆𝑘= ∙, 𝑢𝑖 𝑢𝑖𝜆𝑖=𝜆𝑘

. Since the linear

subspace generated by 𝑢𝑖, with 𝜆𝑖 = 𝜆𝑘, is precisely ker 𝑇 − 𝜆𝑘𝐼 , we obtain that

𝑃𝑘 is nothing but the projection of 𝐻 over ker 𝑇 − 𝜆𝑘𝐼 .

Theorem (spectral resolution of a compact self-adjoint operator):

Let 𝑇 ∈ 𝐿(𝐻) be a compact self-adjoint operator. For every eignevalue 𝜆 let 𝑃𝜆 be

the orthogonal projection on ker 𝑇 − 𝜆𝐼 . Then the family 𝜆𝑃𝜆 𝜆∈𝜎𝑝(𝑇) is summable

in the Banach space 𝐿 𝐻 , and

𝑇 = 𝜆𝑃𝜆.

𝜆∈𝜎𝑝(𝑇)

Moreover, for every 𝜆 ∈ 𝜎𝑝(𝑇)\*0+, the corresponding projection 𝑃𝜆 has finite rank

and these projections are mutually orthogonal, i.e. if λ and 𝜇 are non−equal eigenvalues, then 𝑃𝜆𝑃𝜇 = 𝑃𝜇𝑃𝜆 = 0.

Remark: Now each λ

appears only once.

Compact operators on Hilbert spaces: spectral theorem

Theorem (spectral resolution of a compact normal operator):

Let 𝑇 ∈ 𝐿 𝐻 be a compact normal operator. For every eigenvalue 𝜆, let 𝑃𝜆 be

the orthogonal projection on the invariant subspace ker 𝑇 − 𝜆𝐼 . Then, the family

𝜆𝑃𝜆 𝜆∈𝜎𝑝(𝑇) is summable in the Banach space 𝐿 𝐻 , and

𝑇 = 𝜆𝑃𝜆.𝜆∈𝜎𝑝(𝑇)

Moreover, for every 𝜆 ∈ 𝜎𝑝(𝑇)\*0+, the corresponding projection 𝑃𝜆 has finite

rank, and for every non-equal eigenvalues λ, and 𝜇, we have that

𝑃𝜆𝑃𝜇=𝑃𝜇𝑃𝜆=0.

Corollary: If 𝑇 ∈ 𝐿(𝐻) is a compact normal operator, then 𝑇 is diagonalizable. In

fact,

𝑇 = 𝜆𝑖 ∙, 𝑢𝑖 𝑢𝑖

𝑖

where 𝜆1, 𝜆2 … is the set of non-zero eigenvalues of 𝑇 and 𝑢1, 𝑢2 … is an

orthonormal system of associated eigenvectors.

Recall that 𝑇 ∈ 𝐿(𝐻) is normal ⇔ 𝑇 = 𝑅 + 𝑖𝑆 where 𝑅 and 𝑆 are self-adjoint

operators such that 𝑅𝑆 = 𝑆𝑅.

Compact operators on Hilbert spaces: spectral theorem

Corollary: Let 𝑇 ∈ 𝐿(𝐻) be a compact normal operator. If 𝑇 = 𝜆𝑖 ∙, 𝑢𝑖 𝑢𝑖𝑖 then, for every 𝑦 ∈ 𝐻 and 𝜆 ≠ 0 we have that:

i) If 𝜆 ∉ 𝜆1, 𝜆2 … , then the equation 𝑇 − 𝜆𝐼 𝑥 = 𝑦 has a unique solution for

every 𝑦 ∈ 𝐻. This solution is given by

𝑥 =1

𝜆 ( 𝜆𝑘𝜆≠𝜆𝑘

𝑦,𝑢𝑘

𝜆𝑘−𝜆𝑢𝑘 − 𝑦).

ii) Otherwise, the equation 𝑇 − 𝜆𝐼 𝑥 = 𝑦 has a solution ⟺ 𝑦 ⊥ ker 𝑇 − 𝜆𝐼 . In this is the case, the general solution is given by

𝑥 =1

𝜆 ( 𝜆𝑘𝜆≠𝜆𝑘

𝑦,𝑢𝑘

𝜆𝑘−𝜆𝑢𝑘 − 𝑦) + 𝑧 (𝑧 ∈ ker 𝑇 − 𝜆𝐼 ).

Proof: If 𝑇 − 𝜆𝐼 𝑥 = 𝑦, then 𝑇𝑥 = 𝜆𝑥 + 𝑦, so that

𝑇𝑥, 𝑢𝑘 = 𝜆𝑥 + 𝑦, 𝑢𝑘 = 𝜆 𝑥, 𝑢𝑘 + 𝑦, 𝑢𝑘

Since 𝑇𝑥, 𝑢𝑘 = 𝜆𝑖 𝑥, 𝑢𝑖 𝑢𝑖𝑖 , 𝑢𝑘 = 𝜆𝑘 𝑥, 𝑢𝑘 we obtain that

𝜆 𝑥, 𝑢𝑘 + 𝑦, 𝑢𝑘 = 𝜆𝑘 𝑥, 𝑢𝑘 ,

and hence (𝜆𝑘 − 𝜆) 𝑥, 𝑢𝑘 = 𝑦, 𝑢𝑘 .

Therefore if (𝜆𝑘 − 𝜆)≠ 0, then 𝑥, 𝑢𝑘 =1

𝜆𝑘−𝜆 𝑦, 𝑢𝑘 . Consequenly

𝑥 =1

𝜆(𝑇𝑥 − 𝑦) =

1

𝜆( 𝜆𝑖 𝑥, 𝑢𝑖 𝑢𝑖 − 𝑦) =

1

𝜆 ( 𝜆𝑖𝜆≠𝜆𝑖

𝑦,𝑢𝑖

𝜆𝑖−𝜆𝑢𝑖 − 𝑦).

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