de Investigación
AboutGrupo
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
de Investigación
AboutGrupo
the Mathematical
Foundation of Quantum Mechanics
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
Lecture 2: The mathematical foundations of Quantum Mechanics
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.
CIMPA-MOROCCO
MEKNÈS, September 2014
Operator Theory and The Principles of Quantum Mechanics
From Quantum Mechanics to Functional Analysis
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 Banach space is a complete normed space.
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.
Hilbert spaces
Examples of Banach spaces and Hilbert spaces:
a)
b)
c)
d)
e)
f)
g)
ℝ, ℂ (Hilbert)
Espacios de Banach
ℝ𝑛 , ℂ𝑛 (Hilbert)
Espacios de Hilbert
Sequences spaces 𝑙𝑝 (Hilbert si 𝑝 = 2)
Spaces of continuous functions 𝐶,𝑎, 𝑏Spaces of integrable functions 𝐿𝑝 ,𝑎, 𝑏- (Hilbert if 𝑝 = 2)
Matrices 𝑀𝑛×𝑛 (Hilbert)
Spaces of bounded linear operators: 𝐿 𝑋, 𝑌 , 𝐿(𝐻)
Theorem (Cauchy-Schawarz inequality): If 𝐻 is a linear space equipped with a inner
product ∙,∙ then
𝑢, 𝑣 ≤
𝑢, 𝑢
v
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
normed space with the norm 𝑢 =
𝑢, 𝑢
(𝑢 ∈ 𝐻).
∙,∙ , then 𝐻 is a
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.
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:
Polarization identities: Let 𝐻 be a inner product space over K
1
If K = ℝ, then 𝑢, 𝑣 = ( 𝑢 + 𝑣 2 − 𝑢 − 𝑣 2 ).
If K = ℂ, then 𝑢, 𝑣 =
1
4
4
( 𝑢+𝑣
2
− 𝑢−𝑣
2 )+ 𝑖
4
( 𝑢 + 𝑖𝑣
2
− 𝑢 − 𝑖𝑣
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).
2 ).
Hilbert spaces
Question: Given a normed space 𝐻, howv to know if 𝐻 is an inner product space?
If this is the case, then such a norm needs to satisfy the paralelogram indetity
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
v
𝑢+𝑣
2
+ 𝑢−𝑣
2
= 2( 𝑢
2+
𝑣
2)
(Parallelogram identity)
Therefore we know, for instance that:
𝑣
𝑙𝑝 is Hilbert space ⇔ 𝑝 = 2.
𝑢 −v
𝑢
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
Law of cosines (Euclides): 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 𝑐𝑜𝑠𝜃
𝑏
𝑎
𝜃
𝑐
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
𝑢−𝑣
𝑣
𝜃
Fact : 𝑢 ⊥ 𝑣 ⟺ 𝑐𝑜𝑠𝜃=0 ⟺ 𝑢, 𝑣 =0
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 ⊥⊥
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.
Fact: If 𝑋 is a Banach space then,
𝑋 = 𝑌⨁𝑍 is a topological direct sum ⟺ the subspaces 𝑌 and 𝑍 are closed
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 .
Notation: 𝑃𝑀 𝑢 = 𝑚 ∈ 𝑀: 𝑢 − 𝑚 = 𝑑 𝑢, 𝑀
= best approximation from 𝑢 to 𝑀.
Orthogonal projection theorem: Let 𝐻 be Hilbert space and M a closed subspace.
(Topological direct sum)
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) .
Remmark: Results also know as Hilbert projection theorem
Hilbert spaces: orthogonality
Fact: Every closed subspace of a Hilbert space admits a topological complement.
This is a very important property. Indeed it characterizes the Hilbert spaces
Theorem (Lindestrauss-Tzafriri, 1971): Assume that every closed subspace of a
Banach space 𝑋 is complemented. Then 𝑋 is isomophic to a Hilbert space.
To be complemented means to admit a topological complement
.
Hilbert spaces: orthogonality
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.
Proposición: Every orthogonal family of vectors in 𝐻 is linearly independent.
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.
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
𝑛−1
𝑢𝑛 = 𝑣𝑛 −
𝑗=1
is a orthogonal family in 𝐻.
Morever the family *𝑒𝑛 +𝑛∈ℕ given by 𝑒𝑛 =
𝑢𝑛
𝑢𝑛
𝑢𝑗 , 𝑣𝑛
𝑢𝑗 , 𝑢𝑗
𝑢𝑗
is orthonormal.
Hilbert spaces : summable families
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
𝑖∈𝐽 𝑥𝑖 − 𝑥 < 𝜀. If this is the case we write 𝑥 = 𝑖∈𝐼 𝑥𝑖 .
Proposition: A family of positive real numbers *𝑥𝑖 +𝑖∈𝐼 is summable if and only if the
set * 𝑖∈𝐽 𝑥𝑖 : J ⊆ 𝐼 ; 𝐽 𝑓𝑖𝑛𝑖𝑡𝑜+ is bounded. If this is the case then,
𝑥=
𝑥𝑖 : J ⊆ 𝐼 ; 𝐽 𝑓𝑖𝑛𝑖𝑡𝑜+ .
𝑥𝑖 = sup*
𝑖∈𝐼
𝑖∈𝐽
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,
𝑖∈𝐽 𝑥𝑖 < 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.
1
𝑛
Remark: Apply the Cauchy condition with 𝜀 = . Then we obtain 𝐽 1 such that
𝑖∈𝐽 𝑥𝑖
then,
<
1
𝑛
𝑥𝑖 <
if 𝐽 ∩ 𝐽𝜀 = ∅. Hence, 𝐼0 ≔
1
,
𝑛
∀𝑛, so that 𝑥𝑖 = 0.
𝑛
𝐽 1 is a countable set. Moreover if 𝑖 ∈ 𝐼 ∖ 𝐼0
𝑛
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) ⟺ * 𝑒𝑖 2 +𝑖∈𝐼 is summable en ℝ,
in whose case:
𝑖∈𝐼 𝑒𝑖
2
=
𝑖∈𝐼
𝑒𝑖
2
.
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 = 𝑖=3
𝑖=1 𝑥𝑖 𝑒𝑖 , then
𝑖=3
𝑥=
𝑥, 𝑒𝑖 𝑒𝑖
𝑖=1
The idea is to generalize this fact to an arbitrary 𝐻 to have “coordinates”.
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:
𝑖∈𝐼 𝑓(𝑒𝑖 )𝑒𝑖
2
=
𝑖∈𝐼
𝑓(𝑒𝑖 ) 2 .
Particularly: * 𝑥, 𝑒𝑖 𝑒𝑖 +𝑖∈𝐼 summable in 𝐻 ⟺ * 𝑥, 𝑒𝑖
2+
𝑖∈𝐼
summable ℝ .
Hilbert spaces : summable families
When * 𝑥, 𝑒𝑖
2+
𝑖∈𝐼
is summable in ℝ ? When is 𝑥 =
𝑖∈𝐼
𝑥, 𝑒𝑖 𝑒𝑖 ?
Let 𝐽 ⊆ 𝐼, such that 𝐽 is finite. Then
2
0≤ 𝑥−
𝑥, 𝑒𝑖 𝑒𝑖
= 𝑥−
𝑖∈𝐽
Therefore, 𝑖∈𝐽 𝑥, 𝑒𝑖
Consequently
𝑖∈𝐼
𝑥, 𝑒𝑖. 𝑒𝑖 , 𝑥 −
𝑖∈𝐽
2
𝑢, 𝑒𝑖
This means that * 𝑥, 𝑒𝑖
≤ 𝑥
2
2
2
−
𝑖∈𝐽
𝑥, 𝑒𝑖
2
𝑖∈𝐽
for every 𝐽 ⊆ 𝐼, with 𝐽 finite.
= sup*
2+
𝑖∈𝐼
𝑥, 𝑒𝑖 𝑒𝑖 =⋅⋅⋅= 𝑥
𝑖∈𝐽
𝑢, 𝑒𝑖
2:
J ⊆ 𝐼 ; 𝐽 𝑓𝑖𝑛𝑖𝑡e+ ≤
𝑥
2
is summable in ℝ and hence we have the following:
Theorem (Bessel's inequality): If *𝑒𝑖 +𝑖∈𝐼 is a orthormal family in a Hilbert space 𝐻,
then for every 𝑢 ∈ 𝐻, the family 𝑖∈𝐼 𝑢, 𝑒𝑖 𝑒𝑖 is summable. Moreover
𝑢, 𝑒𝑖 𝑒𝑖 ≤ 𝑢
𝑖∈𝐼
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 = 𝑖∈𝐼 𝑢, 𝑒𝑖 𝑒𝑖
ii) 𝑢, 𝑣 = 𝑖∈𝐼 𝑢, 𝑒𝑖 𝑒𝑖 , 𝑣 (Parseval’s identy)
2
iii) 𝑢 =
.
𝑖∈𝐼 𝑢, 𝑒𝑖
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 =
The other assertions are easy to prove.
𝑖∈𝐼
𝑢, 𝑒𝑖 𝑒𝑖 .
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 ∶
𝛼𝑛 2 < ∞+
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
Theorem: Every Hilbert space 𝐻 has an orthonormal basis. Moreover all the
orthonormal bases of 𝐻 have have the same cardinal.
Definition: The Hilbert space dimension of a Hilbert space 𝐻 is the cardinality of an
orthonormal basis of 𝐻.
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
Definition: Let 𝐻 be a Hilbert space and *𝑒𝑖 +𝑖∈𝐼 a Hilbert basis. For every 𝑢 ∈ 𝐻,
𝑢 = 𝑖∈𝐼 𝑢, 𝑒𝑖 𝑒𝑖 (Fourier expansion)
Fact: The Fourier coefficients * 𝑢, 𝑒𝑖 +𝑖∈𝐼 are the unique family of scalars *𝛼𝑖 +𝑖∈𝐼
such that 𝑢 = 𝑖∈𝐼 𝛼𝑖 𝑒𝑖 vector coordinates in ∞-dim!!
Hilbert spaces : Fourier expansion
Example: Let
be the Lebesgue 𝜎-algebra of −𝜋, 𝜋 and let 𝑚 be the
𝑚
𝑚
Lebesgue measure. Let 𝜇 =
if K = ℝ, and 𝜇 =
if K = ℂ . Then
𝜋
2𝜋
The family 𝑒 𝑖𝑛𝑡 : 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 = 𝑛∈ℤ 𝑓 (𝑛) 𝑒 𝑖𝑛𝑡 (Fourier expansion)
convergence in ∙ 2, where
1 𝜋
𝑓 (𝑛):= ∫−𝜋 𝑓(𝑡)𝑒 −𝑖𝑛𝑡 𝑑𝑡
2𝜋
1
Similarly, the family , cos 𝑛𝑡 , sin(𝑛𝑡) : 𝑛 ∈ ℕ is a Hilbert basis of
2
𝐿2 ( −𝜋, 𝜋 , , 𝜇)ℝ. Therefore, for 𝑓 ∈ 𝐿2 −𝜋, 𝜋 we have
𝑎
𝑓(𝑡) = 0+ ∞
𝑛=1(𝑎𝑛 cos 𝑛𝑡 + 𝑏𝑛 sen 𝑛𝑡 ) (Fourier expansion)
2
Convergence in ∙ 2, where:
1
𝜋
𝑎0 = ∫−𝜋 𝑓(𝑡) 𝑑𝑡 ;
𝜋
1
𝜋
𝑎𝑛 = ∫−𝜋 𝑓(𝑡) cos 𝑛𝑡 𝑑𝑡 ;
𝜋
1
𝜋
𝑏𝑛 = ∫−𝜋 𝑓(𝑡) sen 𝑛𝑡 𝑑𝑡
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 = 𝑖∈𝐼 𝑢, 𝑒𝑖 𝑒𝑖
and
2.
𝑢 =
𝑖∈𝐼 𝑢, 𝑒𝑖
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
isometric isomorphism.
ℂ
→ 𝑙2(ℤ), given by 𝑓 → 𝑓 (𝑛)
n∈ℤ
defines an
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
Let 𝐻 be a Hilbert space over K. Who is 𝐻 ∗ ?
This is to determine the set of continuous
linear functionals 𝑓: 𝐻 → K.
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 = 𝑛𝑖=1 𝛼𝑖 𝑒𝑖 , we have
𝑓 𝑢 =
𝑛
𝑖=1 𝛼𝑖 𝑓(𝑒𝑖 )
= 𝑢, 𝑣
where 𝑣 = 𝑛𝑖=1 𝑓(𝑒𝑖 )𝑒𝑖 . 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 .
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 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 .
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.
Definition: A sequence 𝑢𝑛 , in a Hilbert 𝐻, converges weakly to 𝑢 ∈ 𝐻 (and we
𝜔
write 𝑢𝑛 → 𝑢 ) whenever 𝑢𝑛 , 𝑣 → 𝑢, 𝑣 , for every 𝑣 ∈ 𝐻.
From the continuity of the inner product we deduce the following result.
Corollary: If in a Hilbert space, 𝑢𝑛
∙
𝜔
𝑢 ⇒ 𝑢𝑛 → 𝑢.
The converse does not hold. For instance, in 𝑙2 we have that 𝑒𝑛 , 𝑣 →0 = 0, 𝑣 ,
𝜔
so that 𝑒𝑛 → 0, meanwhile 𝑒𝑛 − 𝑒𝑚 = 2.
Corollary: The (norm)-closed unit ball in a Hilbert space is weakly compact.
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
Consequently, 𝑇 ∗ is continuous and
≤ 𝑇
𝜔 .
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 ∗ = 𝛼𝑇 ∗ (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
Indeed:
𝑇 ∗∗ ≤ 𝑇 ∗ , we obtain that
𝑇∗ = 𝑇 .
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 =
𝑥 = 𝑥1 , 𝑥2 … . . 𝑥𝑛 , … → 𝑇𝑥 = *
∞ 𝑥𝑛
𝑛=1 𝑛 𝑒1 . That is
∞ 𝑥𝑛
𝑛=1 𝑛 , 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.
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 )∗ .
Notation: 𝐿(𝐻) = 𝐿(𝐻, 𝐻).
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 = 𝜆𝑖 𝑒𝑖 ), then 𝑇 ∗ 𝑒𝑖 = 𝜆𝑖 𝑒𝑖 .
ii) If 𝑇 ∈ 𝐿 𝐻 is the orthogonal projection on a closed subspace, then 𝑇 = 𝑇 ∗ .
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 .
Operators in Hilbert spaces: The adjoint operator
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.
Recall that if 𝑇 ∈ 𝐿 𝐻 , then 𝑇 ∗ is the unique operator 𝐿(𝐻) such that
𝑇𝑢, 𝜔 = 𝑢, 𝑇 ∗ 𝜔 , (𝑢, 𝑣 ∈ 𝐻).
Definition: 𝑇 ∈ 𝐿(𝐻) is a self-adjoint operator if 𝑇 = 𝑇 ∗ .
Proposición: Let S, 𝑇 ∈ 𝐿 𝐻 be selft-adjoint operators. Then,
i) 𝑆 + 𝑇 is selft-adjoint.
ii) 𝑆𝑇 selft-adjoint ⟺ 𝑆𝑇 = 𝑇𝑆.
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: 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
+
𝑇−𝑇 ∗
𝑖
.
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.
𝑇 ∈ 𝐿 𝐻 self-adjoint ⟹ 𝑇 normal
𝑇 ∈ 𝐿(𝐻) diagonal ⟹ 𝑇 normal
Proposition: Let 𝑇 ∈ 𝐿 H . Then 𝑇 normal ⟺
𝑇𝑢 = 𝑇 ∗ 𝑢
𝑢∈𝐻 .
Proposition: Let 𝐻 be a complex Hilbert space. Then 𝑇 ∈ 𝐿 𝐻 is normal if, and
only if, its real and imaginary parts commute.
Proposition: If 𝑇 ∈ 𝐿(𝐻) is such that 𝑇 2 = 𝑇 (i.e. 𝑇 is a projection), then,
T normal ⟺ T is a orthogonal projection ⟺ 𝑇 = 1.
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 −1 𝑦.
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
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+
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 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.
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 = ∅.
Fact: Let 𝑇 ∈ 𝐿 𝐻 . By the Banach isomorphism theorem we have that there exists
𝑇 −1 ∈ 𝐿(𝐻) such that 𝑇𝑇 −1 = 𝑇 −1 𝑇 = 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 .
Operators in Hilbert spaces: The spectrum
Theorem: Let 𝐻 be a complex Hilbert space, and let 𝑇 ∈ 𝐿 H . Then, 𝜎 𝑇 is a
non-empty compact subset of ℂ.
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).
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.
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 .
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.
Agreement: From now on, only complex Hilbert spaces will be considered.
Operators in Hilbert spaces: The spectrum
Proposition : If 𝑇 ∈ 𝐿 H then 𝜎 𝑇 ∗ = *λ ∶ 𝜆 ∈ 𝜎 𝑇 + (the conjugate set of 𝜎 𝑇 ).
Proof: 𝑇 − 𝜆𝐼 invertible ⇔ 𝑇 − 𝜆𝐼 𝑅 = 𝑅 𝑇 − 𝜆𝐼 = 𝐼 ⇔ 𝑅∗ 𝑇 − 𝜆𝐼
⇔ 𝑅 ∗ 𝑇 ∗ − 𝜆𝐼 = 𝑅∗ 𝑇 ∗ − 𝜆𝐼 = 𝐼 ⇔ 𝑇 ∗ − 𝜆𝐼 invertible.
∗
= 𝑇 − 𝜆𝐼 ∗ 𝑅 ∗ = 𝐼 ∗
Corollary: If 𝑇 ∈ 𝐿 H is self-adjoint then 𝜎 𝑇 ⊆ ℝ.
Proposition: If 𝑇 ∈ 𝐿 H is normal then,
𝜆 ∈ 𝜎(𝑇) ⟺ there exists 𝑥𝑛 with 𝑥𝑛 = 1 such that
Note: This is equivalent to the following fact:
𝜆 ∈ ℂ\𝜎(𝑇) ⟺ there exists 𝑐 > 0 such that
𝑇 − 𝜆𝐼 𝑥
(𝑇 − 𝜆𝐼)𝑥𝑛 → 0.
≥𝑐 𝑥 ,
Corollary: If 𝑇 ∈ 𝐿 H is normal, then 𝑇 = max 𝜆 : 𝜆 ∈ 𝜎 𝑇 .
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).
𝑥∈𝐻 .
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 = ∫Ω 𝑘 𝑠, 𝑡 𝑥 𝑡 𝑑𝜇(𝑡)
(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.
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.
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).
Compact operators on Hilbert spaces
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.
Notation: Let 𝑋 and 𝑌 be normed spaces. Let denote
𝐾(𝑋, 𝑌)= *𝑇 ∈ 𝐿 𝑋, 𝑌 : 𝑇 is compact+.
Proposition: 𝐾 X, Y is closed whenever 𝑌 is a Banach space.
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 = 𝑙𝑖𝑚𝜋𝑛 𝑇
where 𝜋𝑛 is the orthogonal projection on 𝐿𝑖𝑛 𝑒1 , … , 𝑒𝑛 .
Compact operators on Hilbert spaces
Notation: Let 𝐻 and 𝐾 be Hilbert spaces. Let denote
𝐹 𝐻, 𝐾 = *𝑇 ∈ 𝐿 𝐻, 𝐾 : 𝑇 has finite rank+.
Corollary (Jordan): Let 𝐻 and 𝐾 be Hilbert spaces. Then, 𝐾 𝐻, 𝐾 = 𝐹(𝐻, 𝐾).
Note that the integral operator 𝑇 with kernel 𝑘 is the limit of the sequence of finiterank 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 ∗ − 𝜆𝐼 𝑥 = 𝑦
(c) 𝑇 − 𝜆 𝐼 𝑥 = 0
(d) 𝑇 ∗ − 𝜆𝐼 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)−1 𝑦 meanwhile 𝑥 = (𝑇 − 𝜆𝐼)−1 𝑦.
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 ∗ − 𝜆𝐼
Similarly (b) has a solution ⟺ 𝑦 ∈ 𝑇 ∗ − 𝜆𝐼 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
eigenvalue of 𝑇.
Indeed it is known that if 𝑇 is normal, then
is an
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 − 𝜆1 𝐼 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, 𝜆1 𝑢1 = 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 = 𝑛𝑖=1 𝑢, 𝑢𝑖 𝑢𝑖 + 𝑣 with 𝑣 ∈ 𝐻𝑛+1 . Thus, if 𝑇∕𝐻𝑛+1 = 0 then
𝑇𝑢 =
(this is the case, particularly, if 𝑑𝑖𝑚𝐻<∞).
𝑛
𝑖=1 𝜆𝑖
𝑢, 𝑢𝑖 𝑢𝑖
Compact operators on Hilbert spaces
If the process does not stop, then we obtain a sequence of eigenvalues 𝜆𝑛 → 0.
1
1
Indeed, if 𝜆𝑛 → 𝜆 > 0 then
𝑢𝑛 is bounded so that 𝑇( 𝑢𝑛 )= 𝑢𝑛 has a
𝜆𝑛
𝜆𝑛
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 ⊥.
𝑛
𝑖=1
𝑢, 𝑢𝑖 𝑢𝑖 + 𝑣𝑛 with 𝑣𝑛 ∈ 𝑢1 , 𝑢2 ,…, 𝑢𝑛 ⊥ = 𝐻𝑛+1 , then we obtain that
𝑇𝑣𝑛 = 𝑇∕𝐻𝑛+1 𝑣𝑛 ≤ 𝑇∕𝐻𝑛+1 𝑣𝑛 = 𝜆𝑛+1 𝑣𝑛 ≤ 𝜆𝑛+1 𝑢 → 0,
And hence,
If 𝑢 =
∞
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,
𝑛
𝑇𝑢 =
𝜆𝑖 𝑢, 𝑢𝑖 𝑢𝑖
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
𝑢1 , 𝑢2 … .
real numbers
𝜆1 , 𝜆2 …
and a orthonormal system
Rearranging the above sum, fix 𝑘 and denote 𝑃𝜆𝑘 = 𝜆𝑖=𝜆𝑘 ∙, 𝑢𝑖 𝑢𝑖 . Since the linear
subspace generated by 𝑢𝑖 , with 𝜆𝑖 = 𝜆𝑘 , is precisely ker 𝑇 − 𝜆𝑘 𝐼 , we obtain that
𝑃𝑘 is nothing but the projection of 𝐻 over ker 𝑇 − 𝜆𝑘 𝐼 .
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
Remark: Now each λ
𝑇=
𝜆𝑃𝜆 .
appears only once.
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 𝑃𝜆 𝑃𝜇 = 𝑃𝜇 𝑃𝜆 = 0.
Compact operators on Hilbert spaces: spectral theorem
Recall that 𝑇 ∈ 𝐿(𝐻) is normal ⇔ 𝑇 = 𝑅 + 𝑖𝑆 where 𝑅 and 𝑆 are self-adjoint
operators such that 𝑅𝑆 = 𝑆𝑅.
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 = 𝜆∈𝜎𝑝 (𝑇) 𝜆𝑃𝜆 .
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
𝑃𝜆 𝑃𝜇 =𝑃𝜇 𝑃𝜆 =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
orthonormal system of associated eigenvectors.
𝑢1 , 𝑢2 … is an
Compact operators on Hilbert spaces: spectral theorem
Corollary: Let 𝑇 ∈ 𝐿(𝐻) be a compact normal operator. If 𝑇 = 𝑖 𝜆𝑖 ∙, 𝑢𝑖 𝑢𝑖
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
1
𝜆
𝑥= (
𝑦,𝑢𝑘
𝜆≠𝜆𝑘 𝜆𝑘 𝜆 −𝜆
𝑘
𝑢𝑘 − 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
1
𝜆
𝑥= (
𝑦,𝑢𝑘
𝜆≠𝜆𝑘 𝜆𝑘 𝜆 −𝜆
𝑘
𝑢𝑘 − 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 = 𝑖 𝜆𝑖 𝑥, 𝑢𝑖 𝑢𝑖 , 𝑢𝑘 = 𝜆𝑘 𝑥, 𝑢𝑘 we obtain that
𝜆 𝑥, 𝑢𝑘 + 𝑦, 𝑢𝑘 = 𝜆𝑘 𝑥, 𝑢𝑘 ,
and hence (𝜆𝑘 − 𝜆) 𝑥, 𝑢𝑘 = 𝑦, 𝑢𝑘 .
1
Therefore if (𝜆𝑘 − 𝜆)≠ 0, then 𝑥, 𝑢𝑘 =
𝑦, 𝑢𝑘 . Consequenly
1
𝜆
1
𝜆
𝜆𝑘 −𝜆
𝑥 = (𝑇𝑥 − 𝑦) = ( 𝜆𝑖 𝑥, 𝑢𝑖 𝑢𝑖 − 𝑦) =
1
𝜆
(
𝜆≠𝜆𝑖 𝜆𝑖
𝑦,𝑢𝑖
𝜆𝑖 −𝜆
𝑢𝑖 − 𝑦).