Spectral Theory in Hilbert Spaces [MA5092]1em SS 2013 · Spectral Theory in Hilbert Spaces [MA5092]...

Spectral Theory in Hilbert Spaces [MA5092]

SS 2013

Script by Josias Reppekusbased on lecture by Prof. Dr. Simone Warzel

at Technische Universitï¿œt Mï¿œnchen

4th June 2013

Contents0 Intro – Why spectral theory? 1

0.1 Recap on basics of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Recap on basics of linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

0.2.1 Excursion on Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . 50.2.2 Important notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.3 Notions of convergence in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . 50.3.1 Special features in Hilbert spaces X . . . . . . . . . . . . . . . . . . . . . . 50.3.2 Notions of convergence of linear operators . . . . . . . . . . . . . . . . . . . 6

1 Spectral representation of compact operators 61.1 Further characterisation on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 61.2 Spectral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Hilbert-Schmidt and other Schatten classes . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Convenient characterisation of Hilbert-Schmidt operators . . . . . . . . . . 121.3.2 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Unbounded linear operators 142.1 Adjoint and graph of an operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Calculus of adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Description of the adjoint through its graph . . . . . . . . . . . . . . . . . . 18

2.2 Closed and closable operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Definitions and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Stability of closedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Self-adjoint and normal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Fundamentals of spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Spectrum and resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.2 Properties of the resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Spectra of symmetric and self-adjoint operators . . . . . . . . . . . . . . . . . . . . 272.6 Point, continuous and residual spectrum . . . . . . . . . . . . . . . . . . . . . . . . 30

0 Intro – Why spectral theory?

0 Intro – Why spectral theory?Recall (Linear Algebra). Any self-adjoint (s.a.) n × n-matrix A = A+ is unitarily equivalent toa real valued diagonal matrix, i.e.

A = U diag (λ1, ..., λn)U+. (0.1)

• λl, ..., λn ∈ R eigenvalues of A,

• U = (v1...vn) , vk ∈ Cn, k ∈ [n] orthonormal eigenbasis.

Equivalently

A =

n∑j=1

λjPj , (0.2)

where Pj is the orthogonal projection onto vj.

Note. This provides a full classification of self-adjoint linear operators on Cn.

Question. Is there a natural generalisation of this classification for self-adjoint operators A : H →H on infinite dimensional Hilbert spaces (HS)?

• Why is this interesting?

1. Representation as in (0.1) provide an immediate answer to basic questions about linearequations

Ax = y.

(a) Does it have a solution for every y?(b) Is this solution unique?(c) Does the unique solution depend continuously on y?

2. Representation (0.2) enables functional calculus:

f (A) :=

n∑j=1

f (λj)Pj

is a self-adjoint n× n-matrix for every f : R→ R.3. Spectral theory is an integral part of quantum mechanics.

• Why Hilbert spaces?

1. Most important case for quantum mechanics.2. Classification of linear operators on Banach spaces is more complicated and not com-

pletely understood.

0.1 Recap on basics of Hilbert spacesA Hilbert space H is a complete K-vector space (K ∈ R,C) endowed with a scalar product

〈·, ·〉 : H ×H → K linear in second entry.

Example 0.1 (FA 1). 1. For every measure space (Ω, µ) :

L2 (Ω, µ) :=

f : Ω→ K, f measurable,

ˆ|f |2 dµ <∞

is Hilbert space with scalar product

〈f, g〉 :=

ˆfg dµ

(where we identify functions which differ only on µ-null sets).

1

0 Intro – Why spectral theory? 0.1 Recap on basics of Hilbert spaces

Notation. For µ Lebesgue measure dx, we write L2 (Ω) := L2 (Ω, µ) .

Remark. 1. The space of continuous functions C ([a, b]) equipped with scalar product

〈f, g〉 :=

ˆ b

a

fg dx

is not complete and hence only a pre-Hilbert space.

2. `2 (K) :=

(an)n ∈ KN |∑n |an|

2<∞

is Hilbert space with scalar product

〈a, b〉 :=∑n

anbn.

3. Sobolev space H1 (Ω) = W 1,2 (Ω) equipped with

〈f, g〉H1 :=

ˆfg dx+

ˆDfDg dx

is a Hilbert space.

• Hilbert spaces enable the concept of orthogonality

1. ∀x, y ∈ H :x ⊥ y :⇔ 〈x, y〉 = 0.

2. ∀x ∈ H,Y ∈ H subspace:

x ⊥ Y :⇔ ∀y ∈ Y : 〈x, y〉 = 0.

3. orthogonal subspace to subspace Y ⊆ H :

Y ⊥ := x ∈ H | x ⊥ Y .

Remark. Y ⊥ is closed subspace (exercise).

• A sequences enn∈N is a Hilbert basis (or ONB) of a Hilbert space H, iff

1. 〈en, em〉 = δnm :=

1, n = m

0, else,

2. span en | n ∈ N is dense in H.Remark. Every separable Hilbert space admits an ONB.

• General (non-seperable) Hilbert spaces only admit a total orthonormal system (eα)α∈A (Anot necessarily countable):

1. 〈eα, eβ〉 = δαβ

2. span eα | α ∈ A is dense in H.

Remark. Representation of x ∈ H :

x =∑n

〈en, x〉 en, (convergence in H!)

where the sum only contains countably many non-zero terms. In particular, one has

‖x‖2 =∑α

|〈eα, x〉|2 . (Parseval’sidentity)

2

0 Intro – Why spectral theory? 0.2 Recap on basics of linear operators

0.2 Recap on basics of linear operators• Setting: X,Y normed spaces, D (A) ⊆ X, A : D (A)→ Y linear.

• Notions:

– D (A) is the domain of A (often D (A) = X).– R (A) := Ax | x ∈ D (A) is the range of A.– N (A) := x ∈ D (A) | Ax = 0 is the null space of A.

• A bounded :⇔ A has a finite operator norm ‖A‖ := sup ‖Ax‖ | x ∈ D (A) , ‖x‖ = 1 .

Remark. 1. A bounded ⇔ A continuous ⇔ A continuous at 0.

2. Operator norm renders

L (X,Y ) := A : X → Y | A continuous, linear

into a normed vector space. If Y is a Banach space, so is (L (X,Y ) , ‖·‖). We abbreviateL (X,X) := L (X) .

3. X,Y Hilbert spaces

‖A‖ := sup |〈y,Ax〉| | x ∈ X, y ∈ Y, ‖x‖ = ‖y‖ = 1 .

Example 0.2. 1. Maximal multiplication operator on L2 (Ω, µ)

• measurable function t : Ω→ K,• D (Mt) :=

f ∈ L2 (Ω, µ) | tf ∈ L2 (Ω, µ)

,

• Mtf := tf .

Proposition 0.3. For (X,µ) σ-finite

Mt ∈ L(L2 (Ω, µ)

)⇔ t ∈ L∞ (Ω, µ) .

In this case‖Mt‖ := ‖t‖∞ .

Proof. “⇐” Evidently, D (Mt) = L2 (Ω, t) and ∀f ∈ L2 (Ω, µ)

‖Mtf‖ =

(ˆΩ

|tf | dµ)1/2

≤ ‖t‖∞ ‖f‖2

⇒ ‖Mt‖ ≤ ‖t‖∞ .

“⇒” ∀f ∈ L2 (Ω, µ) : ‖Mtf‖ ≤ ‖Mt‖ ‖f‖ . Since (Ω, µ) is σ-finite,

Ω =⋃n

Ωn, with µ (Ωn) <∞.

Fix n and A > ‖Mt‖ and let fA,n := 1x∈Ωn||t(x)|>A. Then

A2µ (x ∈ Ωn | |t (x)| > A) ≤ˆ|t (x)|2 |fA,n (x)| dµ (x)

= ‖MtfA,n‖2

≤ ‖Mt‖2ˆ|fA,n (x)|2 dµ (x)

= ‖Mt‖2 µ (x ∈ Ωn | |t (x)| > A) ,

hence ∀A > ‖Mt‖ , n:

µ (x ∈ Ωn | |t (x)| > A) = 0

or equivalently t ∈ L∞ (Ω, µ) . Since ‖t‖∞ is infimum of all constants A satisfyingthe previous equality for all n, we also have ‖Mt‖ = ‖t‖∞.

3

0 Intro – Why spectral theory? 0.2 Recap on basics of linear operators

2. Integral operator on L2 (Ω)

• k ∈ L2 (Ω× Ω)

• Kf (x) :=´k (x, y) f (y) dy ∀f ∈ L2 (Ω) = D (K)

‖Kf‖2 =

ˆ|Kf (x)|2 dx

=

ˆ ˆf (y) f (y′)

ˆk (x, y) k (x, y′) dx dy dy′ (Fubini)

≤ˆ ˆ

|f (y)| |f (y′)| ‖k (·, y)‖2 ‖k (·, y′)‖2 dy dy′ (Cauchy-Schwarz)

≤

(√ˆ|f (y)|2 dy

√ˆ ˆ|k (x, y)| dx dy

)2

(Cauchy-Schwarz)

= ‖k‖2L2(Ω×Ω) ‖f‖2L2

⇒ ‖K‖ ≤ ‖k‖L2(Ω×Ω) .

Theorem 0.4 (Riesz-Frï¿œchet representation). Bounded linear functionals on a Hilbert space H(i.e. elements of the dual H ′) can be identified with

H ′ = L (H,K) = 〈g, ·〉 | g ∈ H ∼= H. (0.3)

Definition 0.5. For Hilbert spaces X,Y the (Hilbert) adjoint of A ∈ L (X,Y ) is the uniqueoperator A∗ : Y → X s.t.

∀x ∈ X, y ∈ Y : 〈y,Ax〉 = 〈A∗y, x〉 .

Remark. 1. Using the identification (0.3), A∗ coincides with the notion of adjoint in FunctionalAnalysis I.

2. Properties: ∀A,B ∈ L (X,Y ) :

(a) ‖A∗‖ = ‖A‖,(b) (A∗)

∗= A,

(c) (AB)∗

= B∗A∗.

Definition 0.6. A ∈ L (X) , X Hilbert space, is called self-adjoint (s.a.), iff A = A∗.

Further special classes of operators on Hilbert spaces

Definition 0.7. X Hilbert space.

1. P ∈ L (X) is called orthogonal projection onto R (P ), iff

(a) P = P ∗ (self-adjoint)

(b) P 2 = P (idempotent)

2. A ∈ L (X) is positive, iff ∀x ∈ X : 〈x,Ax〉 ≥ 0.

Definition 0.8. X,Y Hilbert spaces.

1. U ∈ L (X,Y ) is an isometry, iff ∀x ∈ X : ‖Ux‖ = ‖x‖.

2. U ∈ L (X,Y ) is unitary, iff

(a) U is an isometry,

(b) R (U) = Y .

4

0 Intro – Why spectral theory? 0.3 Notions of convergence in Hilbert spaces

Lemma 0.9. For Hilbert spaces X,Y , the following are equivalent.

1. U ∈ L (X,Y ) is unitary.

2. U∗U = 1X and UU∗ = 1Y .

3. U∗ ∈ L (Y,X) is unitary.

Proof. Exercise.

0.2.1 Excursion on Fourier Transformation

An important example of a unitary operator on L2 (Rn) is the Fourier transformation. F :L1 (Rn)→ L∞ (Rn),

f 7→ F1f (k) :=

ˆf (x) e−2πik·x dx.

Properties:

1. ‖F1‖ = 1 (trivial).

2. ∀f, g ∈ C∞c (Rn) :

ˆfg dx =

ˆ(F1f) (F1g) dk. (Plancharel identity)

(try the proof! See: Lieb/Loss: Analysis, AMS 2001)

One may use the Plancharel identity to extend F1 to an isometric linear map F on C∞c (Rn)‖·‖2 =

L2 (Rn) . E.g.

Ff (·) := limn→∞

ˆ|x|≤n

f (x) e−2πix(·) dx. (L2-convergence!)

One can check〈g, f〉 = 〈Fg,Ff〉 ∀f, g ∈ L2 (Rn) .

The Fourier transformation is surjective, i.e. R (F) = L2 (Rn) since the adjoint F∗ is the inverse

F∗f (·) =

ˆf (x) e2πik(·) dxk.

Hence it is unitary.

0.2.2 Important notions

Definition 0.10. A,B ∈ L (X) are unitarily equivalent, iff ∃unitary U : A = U∗BU.

0.3 Notions of convergence in Hilbert spacesSetting. X normed space, xnn ⊆ X sequence.

• xn → x :⇔ limn→∞ ‖x− xn‖ = 0

• xnw− x :⇔ limn→∞ f (xn)→ f (x) ∀f ∈ X ′

Remark. xn → x ⇒ xnw− x (in general 6⇐).

0.3.1 Special features in Hilbert spaces X

Since X ′ can be identified with X, the notion of weak* convergence coincides with weak conver-gence.

Theorem 0.11 (Consequence of Banach-Alaoglu). Every bounded sequence xnn in a Hilbertspace has a weakly convergent subsequence.

5

1 Spectral representation of compact operators

0.3.2 Notions of convergence of linear operators

Setting. X,Y normed spaces, A,An ∈ L (X,Y ).

• An norm convergent to A or An → A, iff limn→∞ ‖A−An‖ = 0.

• An strongly convergent to A or Ans−→ A, iff ∀x ∈ X : limn→∞ ‖Ax−Anx‖ = 0.

For X,Y Hilbert spaces

• An weakly convergent to A or Anw−→ A, iff ∀x ∈ X, y ∈ Y : limn→∞ 〈y,Anx〉 = 〈y,Ax〉.

Remark. An → A ⇒ Ans−→ A ⇒ An

w−→ A (in general 6⇐).

1 Spectral representation of compact operatorsSetting. X,Y Banach spaces.

Definition 1.1. • A ∈ L (X,Y ) is compact (cpt.) :⇔ A(BX)is compact in Y .

[M ⊆ Y is compact :⇔ Every sequence in M has a subsequence converging in M .]

• K (X,Y ) := A ∈ L (X,Y ) | A compact

Elementary properties of compact operators

1. K (X,Y ) is a closed subspace of L (X,Y ) (wrt. operator norm).

2. A ∈ K (X,Y ) ⇔ ∀xnn ⊆ X bounded: ∃ xnkk subsequence: Axnk converges in Y .

3. Limits of sequences of finite range operators are compact.

4. A ∈ K (X,Y ) ⇔ A∗ ∈ K (Y,X) (Schauder’s theorem, FA I)

1.1 Further characterisation of compact operators on Hilbert spacesProposition 1.2. X Hilbert space, Y Banach space.

1. A ∈ K (X,Y ) ⇔ ∀xnw− x : Axn → Ax.

2. A ∈ K (X,Y ) ⇒ N (A)⊥,R (A) are separable.

Proof. 1. “⇒” Wlog. xnw− 0.

(a) ∀F ∈ Y ′ : F (Axn) = (A∗F ) (xn)→ 0.(b) Since xnn bounded, A compact, ∃ xnkk subsequence: Axnk → A0 = 0.

(a),(b) Axnw− 0 ∧Axnk → 0 ⇒ Axn → 0

“⇐” Let xnn be bounded. Then ∃ xnkk :

xkkw− x ⇒ Axnk → Ax.

2. Let eαα∈A be a total orthonormal system in N (A)⊥ and αnn ⊆ A : αn 6= αm ∀n 6= m.

(a) eαnw− 0, A compact ⇒ Aeαn → 0.

(b) ∀ε > 0 : α ∈ A | ‖Aeα‖ > ε is finite by (a).

Since Aeα 6= 0 by definition, this implies that A is countable. The set Aeα | α ∈ A is densein R (A), so R (A) is separable as well.

Theorem 1.3. Let X be a Hilbert space and A ∈ L (X,Y ).

6

1 Spectral representation of compact operators 1.1 Further characterisation on Hilbert spaces

1. If A ∈ K (X,Y ) and Pnn is a monotone increasing sequence, i.e. Pn ≤ Pn+1 ∀n, of finitedimensional orthogonal projections on X, s.t. Pn

s−→ 1X , then ‖A−APn‖ → 0.

2. A ∈ K (X) ⇔ A is norm limit of bounded operators Ann of finite range.

Proof. 1. Pick xn ∈ N (Pn) , ‖xn‖ ≤ 1 and

‖Axn‖ = ‖A (1− Pn)xn‖ ≥1

2‖A (1− Pn)‖ . (1.1)

Then xnw− 0, since ∀x ∈ X :

〈x, xn〉 = 〈x, (1− Pn)xn〉=⟨(1− Pn)

∗x, xn

⟩= 〈(1− Pn)x, xn〉 → 0.

Since A is compact, this implies Axn → 0. Consequently, by (1.1)

‖A−APn‖ = ‖A (1− Pn)‖ ≤ 2 ‖Axn‖ → 0.

2. It remains to show “⇒”: Since N (A)⊥ is separable, there is an ONB enn of N (A)

⊥. Thesequence of finite dimensional orthogonal projections

Pnx :=

n∑k=1

〈ek, x〉 ek

is monotone increasing and Pns−→ 1N (A)⊥ . By 1., the compact operator A := A|N (A)⊥ :

N (A)⊥ → X satisfies ∥∥∥A− APn∥∥∥→ 0.

Let P be the orthogonal projection onto N (A)⊥. Then∥∥∥A− APn∥∥∥ =

∥∥∥AP − APn∥∥∥ =∥∥∥A− APn∥∥∥→ 0.

Corollary 1.4. Let X,Y Hilbert spaces, A ∈ L (X,Y ) . Then the following are equivalent.

1. A ∈ K (X,Y )

2. Aen → 0 ∀ orthonormal enn ⊆ X.

3. 〈fn, Aen〉 → 0 ∀ orthonormal enn ⊆ X, fnn ⊆ Y .

Proof. 1.⇒ 2. From enw− 0 and A compact.

2.⇒ 1. Pick enn s.t. ‖e1‖ = 1, ‖Ae1‖ ≥ ‖A‖2 and (for n ≥ 1): en+1 ⊆ span e1, ..., en⊥ s.t.‖en+1‖ = 1 and

‖Aen+1‖ = ‖A (1− Pn) en+1‖ ≥‖A (1− Pn)‖

2, Pnx =

n∑k=1

〈ekx〉 ek. (1.2)

Then,‖A−APn‖ = ‖A (1− Pn)‖ ≤ 2 ‖Aen+1‖ → 0,

i.e. A is norm of a limit of bounded operators APn of finite dimensional range.

1.⇔ 3. Exercise.

7

1 Spectral representation of compact operators 1.2 Spectral representation

1.2 Spectral representation for compact operatorsBasic observation:

Lemma 1.5. Let A ∈ K (X) self-adjoint on a Hilbert space X.

1. A has at least one eigenvalue λ ∈ ±‖A‖, i.e.

∃ϕ ∈ X\ 0 : Aϕ = λϕ. (1.3)

2. For ϕ as in (1.3), set X⊥ := ϕ⊥. Then A|X⊥ is self-adjoint and compact in X⊥.

Proof. 1. Since‖A‖ = sup |〈x,Ax〉| | x ∈ X, ‖x‖ = 1 , (Exercise)

there is xnn , ‖xn‖ = 1 s.t. |〈xn, Axn〉| → ‖A‖ .

(a) Since xnn is bounded and X is a Hilbert space, by passing to a subsequence, we mayassume wlog. xn

w− x and ‖x‖ ≤ 1.(b) Since 〈xn, Axn〉 ∈ R (A self-adjoint)

limn→∞

〈xn, Axn〉 = ±‖A‖ .

(c) Since A is compact, Axn → Ax and

‖Ax‖︸︷︷︸≤‖A‖‖x‖

= limn→∞

‖Axn‖︸︷︷︸≥|〈xn,Axn〉|

≥ limn→∞

|〈xn, Axn〉| = ‖A‖ .

⇒ ‖x‖ = 1, ‖Ax‖ = ‖A‖. By (a)

〈x,Ax〉 = limn→∞

〈xn, Axn〉 = ±‖A‖ .

Conclusion.

‖Ax∓ ‖A‖x‖2 = ‖Ax‖2 ∓ 2 Re 〈Ax, ‖A‖x〉+ ‖‖A‖x‖2

≤ ‖A‖2 − 2 ‖A‖2 + ‖A‖2

= 0.

Hence Ax = ±‖A‖x.

2. For x ∈ X⊥ :

〈Ax,ϕ〉 = 〈x,Aϕ〉 (1.3)= λ 〈x, ϕ〉 = 0 ⇒ Ax ∈ X⊥.

Hence, A|X⊥ ∈ L (X⊥). Evidently A|X⊥ is self-adjoint and compact.

Theorem 1.6 (Spectral theorem for compact self-adjoint operators). Let A ∈ K (X) self-adjointon a Hilbert space X. Then there are (at most) countably many λn ∈ R with |λ1| ≥ |λ2| ≥ ... andlimn→∞ λn = 0 (in the infinite dimensional case) and orthonormal vectors ϕn ∈ X s.t.

Ax =∑n

λn 〈ϕn, x〉ϕn ∀x ∈ X. (1.4)

In particular Aϕn = λnϕn.In case X is separable, for any ONB ψnn of N (A) : ψnn ∪ ϕnn is ONB of X.

Remark (Alternative formulation). ∃ pairwise disjoint µn ∈ R with |µn+1| ≤ |µn| , (µn → 0 ininfinite case) and pairwise orthogonal finite dimensional projections Pn, i.e. PnPm = δnmPn s.t.

A =∑n

µnPn,

there the sum converges in operator norm.For a proof, group terms with the same λn and define orthogonal projections onto the span of

the ϕn corresponding to the same λn. The norm convergence follows from:

8


Lemma 1.7. Let µnn be a null sequence and Pnn pairwise orthogonal finite dimensionalprojections on a Hilbert space. Then

∑n µnPn converges in operator norm.

Proof. The partial sums Sn :=∑nk=1 µkPk are a Cauchy sequence w.r.t. operator norm, i.e.

∀n > m

‖Sn − Sm‖ =

∥∥∥∥∥n∑

k=m+1

µkPk︸︷︷︸=:Y

∥∥∥∥∥ = ‖Y ∗Y︸︷︷︸=

n∑k,k′=m+1

µk′µk Pk′Pk︸︷︷︸Pkδkk′

‖12

=

∥∥∥∥∥n∑

k=m+1

|µk|2 Pk

∥∥∥∥∥12

≤

(sup

k∈m+1,...,n|µk|2

)1/2

→ 0.

Remark. The theorem remains true for normal A ∈ K (X).

Proof. By Lemma 1.7: ∃ϕ1 ∈ X\ 0 :

Aϕ1 = ±‖A‖ϕ1.

Set X1 := ϕ1⊥ and iterate the application of Lemma 1.7, first to

A|X1: X1 → X1,

then to A|X2 , X2 := span ϕ1, ϕ2⊥ , etc.

Case 1. Xn = 0 for some n: span ϕ1, ..., ϕn = X (finite dimensional case).

Case 2. Xn 6= 0 for all n: We hence construct a sequence λnn ⊆ R and orthonormal ϕnns.t.

Aϕn = λnϕn.

Hence for Xn := span ϕ1, ...., ϕn⊥ :

• ‖A|Xn‖ = |λn+1| = ‖Aϕn+1‖ → 0, since ϕnw− 0 and A is compact.

• Ax = 0 for all x ⊥ span ϕk | k ∈ N, |λk| > 0, i.e. N (A)⊥

= span ϕk | k ∈ N, |λk| > 0.

The representation (1.4) follows from the representation x =∑〈ϕn, x〉ϕn for any x ∈ N (A)

⊥

with ϕnn any ONB in N (A)⊥. In particular for the ONB constructed above:

Ax =∑n

〈ϕn, x〉Aϕn =∑n

λn 〈ϕn, x〉ϕn.

For positive compact operators A ≥ 0, the eigenvalues are non-negative, since

λn = 〈ϕn, Aϕn〉 ≥ 0.

As an immediate consequence we can define:

Definition 1.8. For A ≥ 0 compact and any k ∈ N, the operator

A1/k :=

∑n

µ1/kn Pn

is called the k-th root of A.

9


Remark. We have(A1/k

)k= A and S = A1/k is the only operator satisfying Sk = A, since

S =∑n

τnQninduction=====⇒

on kSk =

∑n

τknQn!=∑m

µmPm

====⇒exercise

µn = τkn , Qn = Pn ∀n.

Theorem 1.9 (Spectral representation for compact operators). Let X,Y be Hilbert spaces andA ∈ K (X,Y ). Then there are at most countably many sn > 0 (with sn → 0 in the infinite case),orthonormal ϕnn in X and ψn in Y s.t.

Ax =∑n

sn 〈ϕn, x〉ψn ∀x ∈ X,

A∗y =∑n

sn 〈ψn, y〉ϕn ∀y ∈ Y.

Moreover, snn are the non-zero eigenvalues of |A| := (A∗A)1/2. The latter coincide with the

non-zero eigenvalues of |A∗| = (AA∗)1/2. The corresponding orthonormal eigenvectors of |A| are

ϕnn and ψnn for |A∗| respectively.

Remark. 1. The subsequent proof also shows that ∀x ∈ X, y ∈ Y :

‖Ax‖ = ‖|A|x‖ , ‖A∗y‖ = ‖|A∗| y‖ .

2. Simple corollary: A ∈ K (X,Y ) is norm limit of finite range operators

Anx :=

n∑k=1

xk 〈ϕk, x〉ψk.

Proof. Note that A∗A is self-adjoint, positive, and compact. Therefore, there are s2n > 0 and

orthonormal ϕnn :

A∗Ax =∑n

s2n 〈ϕn, x〉ϕn ∀x ∈ X

(Note:s2n

nare the non-zero eigenvalues and ϕnn the corresponding orthonormal eigenvectors

of A∗A). Let

ψn :=1

snAϕn, sn positive root of s2

n.

Then ∀n,m, x ∈ X :

1. 〈ψn, ψm〉 = 1snsm

〈Aϕn, Aϕm〉 = 1snsm

〈A∗Aϕn, ϕm〉 =s2nsnsm

δn,m = δn,m.

2. AA∗ψn = 1snAA∗Aϕn =

s2nsnAϕn = s2

nψn.

3. ‖Ax‖2 = 〈Ax,Ax〉 = 〈A∗Ax, x〉 =⟨|A|2 x, x

⟩= 〈|A|x, |A|x〉 = ‖|A|x‖2 .

Hence N (A) = N (A∗A). Let P be the orthogonal projection on N (A)⊥. Use

Px =∑n

〈ϕn, x〉ϕn ∀x ∈ X

to showAx = APx =

∑n

〈ϕn, x〉Aϕn =∑n

sn 〈ϕn, x〉ψn ∀x ∈ X.

For A∗ proceed analogously using ϕn = 1s2nA∗Aϕn = 1

snA∗ψn.

Definition 1.10. For A ∈ K (X,Y ) , X, Y Hilbert space, the non-zero eigenvalues sn = sn (A) of|A| = (A∗A)

1/2 are called the singular values of A.

10

1 Spectral representation of compact operators 1.3 Hilbert-Schmidt and other Schatten classes

Theorem 1.11 (Min-Max principle). Let A ∈ K (X,Y ) , X, Y Hilbert spaces and s1 (A) > s2 (A) ≥... the ordered sequence of singular values of A. Then)

s1 (A) = sup ‖Ax‖ | x ∈ X, ‖x‖ = 1 ,

sn+1 (A) = infx1,...,xn∈X

sup‖Ax‖ | x ∈ X, ‖x‖ = 1, x ∈ span x1, ..., xn⊥

, n ≥ 1. (1.5)

Proof. Since s1 (A) is the maximal eigenvalue of |A|, we have s1 (A) = ‖|A|‖ = ‖A‖ .For a proof of (1.5) use spectral representation 1.9

Ax =∑n

sn 〈ϕn, x〉ψn ∀x ∈ X.

Hence for x ∈ span ϕ1, ..., ϕn⊥:

‖Ax‖2 =

∥∥∥∥∥∞∑

k=n+1

sk 〈ϕk, x〉ψk

∥∥∥∥∥2

=

∞∑k=n+1

s2k |〈ϕk, x〉|

2

≤ s2n+1

∞∑k=n+1

|〈ϕk, x〉|2

≤ s2n+1 ‖x‖

2. (Bessel inequality)

⇒ sn+1 (A) ≥ r.h.s. of (1.5).Now let x1, ..., xn ∈ X arbitrary. Pick x ∈ span ϕ1, ..., ϕn+1 , ‖x‖ = 1 s.t. x ∈ span x1, ..., xn⊥.

Then

‖Ax‖2 =

∥∥∥∥∥n+1∑k=1

sk 〈ϕk, x〉ψk

∥∥∥∥∥2

=

n+1∑k=1

s2k |〈ϕk, x〉|

2

≥ s2n+1

n+1∑k=1

|〈ϕk, x〉|2

= s2n+1 ‖x‖

2. (Parseval identity)

⇒ sn+1 (A) ≤ r.h.s. of (1.5).

1.3 A glance at Hilbert-Schmidt operators and other Schatten classesDefinition 1.12. An operator A ∈ L (X,Y ) on Hilbert spaces X,Y is a Hilbert-Schmidt operator,iff there is a total orthonormal system eα | α ∈ A such that∑

α∈A‖Aeα‖2 <∞.

Lemma 1.13. Let X,Y Hilbert spaces. A ∈ L (X,Y ) is Hilbert-Schmidt, iff A∗ is Hilbert-Schmidt.In this case for any total orthonormal system eα | α ∈ A in X and fβ | β ∈ B in Y

‖A‖2 ≤∑α∈A‖Aeα‖2 =

∑β∈B

‖A∗fβ‖2 .

11


Proof. A is Hilbert-Schmidt ⇒ ∃ total orthonormal system eα | α ∈ A :∑α ‖Aeα‖

2<∞.

Let fβ | β ∈ B be a total orthonormal system on Y . Then∑β∈B

‖A∗fβ‖2 =∑β∈B

∑α∈A|〈eα, A∗fβ〉|2 (Parseval identity)

=∑α∈A

∑β∈B

|〈Aeα, fβ〉|2

=∑α∈A‖Aeα‖2 <∞ (Parseval identity)

⇒ A∗ is Hilbert-Schmidt.Since we may change the roles of A and A∗ in the above calculation, this implies the equivalence

and the independence of the series of the total orthonormal system.∀ε > 0 : ∃xε ∈ X, ‖xε‖ = 1 :

‖A‖ ≤ ‖Axε‖+ ε.

We may always choose xε as an element of some total orthonormal system eα | α ∈ A . Hence

‖A‖ ≤ ‖Axε‖+ ε ≤

(∑α

‖Aeα‖2)1/2

+ ε.

Definition 1.14. For a Hilbert-Schmidt operator A the value

‖A‖HS :=

(∑α

‖Aeα‖2)1/2

is called the Hilbert-Schmidt norm.

Remark. The Hilbert-Schmidt norm is a norm of the vector space of Hilbert-Schmidt operators(see exercise). It renders this space into a Banach space.

1.3.1 Convenient characterisation of Hilbert-Schmidt operators

Theorem 1.15. Let X,Y Hilbert spaces.

1. Every Hilbert-Schmidt operator is compact.

2. A ∈ K (X,Y ) is Hilbert-Schmidt, iff its singular values are square-summable, i.e.∑n

sn (A)2<∞.

In this case

‖A‖HS =

(∑n

sn (A)2

)1/2

.

Proof. 1. Let enn be an orthonormal sequence. Since∑n ‖Aen‖

2<∞, one has ‖Aen‖ → 0.

Hence A is compact by Corollary 1.4.

2. From the spectral representation 1.9 of A ∈ K (X,Y )

Ax =∑n

sn (A) 〈ϕn, x〉ψn.

Let eα | α ∈ A be a total orthonormal system containing ϕnn.

⇒ Aeα =

sn (A)ψn, if eα ∈ ϕnn0, if eα /∈ span ϕnn

.

⇒∑α

‖Aeα‖2 =∑n

sn (A) <∞.

12


Conversely, if∑n sn (A) <∞, then for the above total orthonormal system eα | α ∈ A we

have ∑n

‖Aeα‖2 =∑n

sn (A)2<∞.

Hence A is Hilbert-Schmidt.

Example 1.16 (Hilbert-Schmidt integral operators). Let (Ωj , µj) , j = 1, 2 measure spaces, andconsider k ∈ L2 (Ω1 × Ω2, µ1 × µ2) . Since k (x, ·) ∈ L2 (Ω2, µ2) for µ1-almost every x ∈ Ω1, theintegral

(Kf) (x) :=

ˆΩ2

k (x, y) f (y) µ2 (dy) (1.6)

is well defined for every f ∈ L2 (Ω2, µ2) and µ1-almost every x ∈ Ω1. It defines a measurablefunction on Ω1 with the property that

|Kf (x)| ≤ ‖f‖L2(Ω2,µ2)

√ˆ|k (x, y)| µ2 (dy)

for µ1-almost every x ∈ Ω1. Hence Kf ∈ L2 (Ω1, µ1) and

‖Kf‖L2(Ω1,µ1) ≤ ‖f‖L2(Ω2,µ2) ‖k‖L2(Ω1×Ω2,µ1×µ2) .

⇒ K ∈ L(L2 (Ω2, µ2) , L2 (Ω1, µ1)

).

We will show that K is Hilbert-Schmidt with

‖K‖HS = ‖k‖L2(Ω1×Ω2,µ1×µ2) =

(ˆΩ1×Ω2

|k (x, y)|2 µ1 (dx) µ2 (dy)

)1/2

.

Let eα | α ∈ A be a total orthonormal system in L2 (Ω, µ2).

‖Keα‖2 =

ˆΩ1

∣∣∣∣ˆΩ2

k (x, y) eα (y) µ2 (dy)

∣∣∣∣ µ1 (dx)

=

ˆΩ1

∑α

∣∣∣⟨k (x, ·), eα⟩∣∣∣2 µ1 (dx)

=

ˆΩ1

‖k (x, ·)‖L2(Ω2,µ2) µ1 (dx) (Parseval identity)

=

ˆΩ1×Ω2

|k (x, y)|2 µ1 (dx) µ2 (dy) <∞. (Fubini-Tonelli)

Remark. In fact, all Hilbert-Schmidt operators on L2 (Ω, µ) are of this form.

Theorem 1.17. An operator K ∈ L(L2 (Ω2, µ2) , L2 (Ω1, µ1)

)is a Hilbert-Schmidt operator, iff

there is k ∈ L2 (Ω1 × Ω2, µ1 × µ2) such that (1.6) holds. In this case

‖K‖HS = ‖k‖L2(Ω1×Ω2,µ1×µ2) .

Proof. “⇐” See example 1.16.

“⇒” Let eα | α ∈ A be a total orthonormal system in L2 (Ω2, µ2). Set

kα (x, y) := (Keα) (x) eα (y).

Then

‖kα‖L2(Ω1×Ω2,µ1×µ2) = ‖Keα‖L2(Ω1,µ1) ,

〈kα, kβ〉L2(Ω1×Ω2,µ1×µ2) = 〈Keα,Keβ〉L2(Ω1,µ1) 〈eα, eβ〉L2(Ω2,µ2) = 0, if α 6= β.

13

2 Unbounded linear operators

Hence ∑α

kα (x, y) =: k (x, y)

converges in L2 (Ω1 × Ω2, µ1 × µ2) (not pointwise!).

The operator K ∈ L(L2 (Ω2, µ2) , L2 (Ω1, µ1)

)defined by(

Kf)

(x) :=

ˆΩ2

k (x, y) f (y) µ2 (dy)

is Hilbert-Schmidt (see example 1.16). Moreover, for all α ∈ A :(Keα

)(x) =

ˆΩ2

k (x, y) eα µ2 (dy)

=∑α′

(Keα′) (x) 〈eα′ , eα〉L2(Ω2,µ2)︸︷︷︸=0 if α6=α′

= (Keα) (x) .

1.3.2 Further topics

1. For X,Y Hilbert spaces, p > 0, one defines

Sp (X,Y ) :=

A ∈ K (X,Y ) |

∑n

sn (A)p<∞

. (Schatten-poperators)

Special cases:

p = 2 : Hilbert-Schmidt operators.

p = 1 : trace class operators.[In caseX = Y for these operators, one may define the trace trA =

∑α 〈eα, Aeα〉 ,

cf. exercise]

2. For p ≥ 1, Sp (X,Y ) forms a Banach space equipped with(∑n

sn (A)p

)1/p

=: ‖A‖Sp.

For p = 2, this is a Hilbert space [with scalar product 〈A,B〉 = trA∗B].

2 Unbounded linear operatorsX,Y Hilbert spaces, A : D (A)→ Y,D (A) ⊆ X subspace.

Recall. A bounded ⇔ ‖A‖ = sup ‖Ax‖Y | x ∈ D (A) , ‖x‖X = 1 <∞.

Example 2.1 (Unbounded operators).

1. Maximal multiplication operator (of class 2) Mtf = tf, t /∈ L∞ (Ω, µ) and

D (Mt) =f ∈ L2 (Ω, µ) | tf ∈ L2 (Ω, µ)

.

2. Differentiation operator D0 : C∞c (I)→ L2 (I) , D0f := f ′ with I = (a, b) ⊆ R.Maximal differentiation operator Dmax : W 1,2 (I)→ L2 (I) , Dmaxf

′.

Note. Since C∞c (I) ⊆W 1,2 (I), the operator Dmax is an extension of D0.

Notation. D0 ⊆ Dmax.

14

2 Unbounded linear operators 2.1 Adjoint and graph of an operator

2.1 Adjoint and graph of an operatorRecall. A ∈ L (X,Y ) , X, Y Hilbert spaces. The adjoint A∗ ∈ L (Y,X) is uniquely determined by

〈y,Ax〉 = 〈A∗y, x〉 ∀x ∈ X, y ∈ Y.

[Namely: A∗y is the y∗ ∈ X that corresponds to the continuous linear functional X 3 x 7→ 〈x,Ax〉].

Definition 2.2. Take X,Y Hilbert spaces, and linear operators

A : D (A)→ Y, D (A) ⊆ X, B : D (B)→ X, D (B) ⊆ Y.

We call A and B formally adjoint, iff

〈y,Ax〉 = 〈By, x〉 ∀x ∈ D (A) , y ∈ D (B) . (2.1)

Observations. 1. If D (A) is not dense in X, then any operator B = B+C with C : D (B)→D (A)

⊥ is also formally adjoint to A, if B was. ⇒ A does not uniquely determine B.

2. If D (A) is dense in X and B1, B2 are formally adjoint to A, then

〈(B1 −B2) y, x〉 = 〈B1y, x〉−〈B2y, x〉 = 〈y,Ax〉−〈y,Ax〉 = 0 ∀x ∈ D (A) , y ∈ D (B1)∩D (B2) .

The operator B : D (B)→ X with

• D (B) = D (B1) +D (B2) := y ∈ Y | ∃y1 ∈ D (B1) , y2 ∈ D (B2) : y = y1 + y2 .• By = B1y1 +B2y2

is well defined as a linear operator. It is an extension of B1 and B2 and formally adjoint toA,

〈By, x〉 = 〈B1y1, x〉+ 〈B2y2, x〉 = 〈y1 + y2, Ax〉 = 〈y,Ax〉 ∀x ∈ D (A) , y ∈ D (B) . (2.1)

This suggests to define the adjoint of A as the unique maximally defined operator which isformally adjoint to A.

Definition 2.3. Let A : D (A) → Y,D (A) ⊆ X be a densely defined linear operator on Hilbertspaces X,Y . the operator A∗ : D (A∗)→ X with

• D (A∗) := y ∈ Y | ∃y∗ ∈ X : 〈y∗, x〉 = 〈y,Ax〉 ∀x ∈ D (A),

• A∗y = y∗

is called the adjoint of A.

Remark. 1. y∗ is the unique element in X that corresponds to the continuous linear functionalD (A)→ K, x 7→ 〈y,Ax〉 .

2. In case A : X → Y is bounded, we have D (A∗) = Y and the above definition coincides withthe old definition.

Example 2.4. 1. Differentiation operator D0 : C∞c (I) → L2 (I) , D0f = f ′. It’s adjoint is−Dmax : W 1,2 (I)→ L2 (I) ,−Dmaxf = −f ′.Note. For f ∈ C∞c (I) , g ∈ W 1,2 (I) : 〈g, f ′〉 =

´Ig (x)f ′ (x) dx = −

´Ig′ (x)f (x) dx =

〈−g′, f〉 .

2. Densely defined operator whose adjoint is only defined on 0 :

Pick sequences (nk,l)l ∈ NN, k ∈ N such that:

(a) nk,l 6= nj,m ∀ (k, l) 6= (j,m).

(b)⋃k∈N nk,l | l ∈ N = N.

15


Set A : D (A)→ l2 (N) with D (A) = l20 (N) [finite sequences in l2 (N)]:

Ax :=

( ∞∑l=1

xnk,l

)k∈N

.

Then D (A∗) = 0, since by formal adjointness ∀y ∈ D (A∗) , x ∈ D (A) :∑k

∑l

ykxnk,l = 〈y,Ax〉 = 〈A∗y, x〉 =∑n

(A∗y)nxn(b)=∑k,l

(A∗y)nk,lxnk,l

(all sums are finite). Hence, for x = enk,l [i.e. xm =

1, m = nk,l

0, else]:

yk = (A∗y)nk,l ∀k, l ∈ N.

⇒ y = 0.

Theorem 2.5. Let A : D (A) → Y,D (A) ⊆ X be a densely defined operator on Hilbert spacesX,Y .

1. If A∗ is densely defined, then (A∗)∗

=: A∗∗ exists and A ⊆ A∗∗ (i.e. A∗∗ is an extension ofA).

2. N (A∗) = R (A)⊥,R (A) = N (A∗)

⊥.

3. A is bounded, iff A∗ ∈ L (X,Y ). In this case ‖A‖ = ‖A∗‖.

4. If A is bounded, then A∗∗ is the unique extension of A to X.

Proof. 1. Obvious from the above.

2. Recall N (A∗) = y ∈ D (A∗) | A∗y = 0. Since D (A) is dense in X, we have

y ∈ N (A∗) ⇔ ∀x ∈ D (A) : 〈y,Ax〉 = 〈A∗y, x〉 = 0

⇔ y ∈ R (A)⊥.

Since(M⊥

)⊥= M for any subspace M of a Hilbert space, this implies R (A) = N (A)

⊥.

3. If A is bounded, then ∀x ∈ D (A) , y ∈ Y :

|〈y,Ax〉| ≤ ‖A‖ ‖x‖ ‖y‖ .

Hence, D (A) 3 x 7→ 〈y,Ax〉 is a continuous linear functional for all y ∈ Y , so D (A∗) = Y .

In this case,

‖A∗‖ = sup |〈A∗y, x〉| | x ∈ X, y ∈ Y, ‖x‖ = ‖y‖ = 1= sup |〈A∗y, x〉| | x ∈ D (A) , y ∈ Y, ‖x‖ = ‖y‖ = 1 (D (A)is dense in X)= sup |〈y,Ax〉| | x ∈ D (A) , y ∈ Y, ‖x‖ = ‖y‖ = 1= ‖A‖ .

Conversely, if A∗ ∈ L (Y,X), then A∗∗ ∈ L (X,Y ) and hence its restriction A is also bounded.

4. From 1.: A ⊆ A∗∗, from 3.: A∗∗ ∈ L (X,Y ). Since A is densely defined, this continuousextension is unique.

16


2.1.1 Calculus of adjoint operators

Theorem 2.6. X,Y, Z Hilbert spaces, A1 : D (A1) → Y,D (A1) ⊆ X dense in X, A2 : D (A2) →Z,D (A2) ⊆ Y dense in Y .

1. If A2A1 is densely defined, then (A2A1)∗ ⊇ A∗1A∗2.

2. If A2 ∈ L (Y,Z), then (A2A1)∗

= A∗1A∗2.

3. If A1 is injective and A−11 ∈ L (Y,X), then A2A1 is densely defined and (A2A1)

∗= A∗1A

∗2.

Remark. In general there is no equality in 1., see exercise!

Proof. 1. It is sufficient to show that A∗1A∗2 is formally adjoint to A2A1.

∀x ∈ D (A2A1) = x ∈ D (A1) | A1x ∈ D (A1A2) , z ∈ D (A∗1A∗2) = z ∈ D (A∗2) | A∗2z ∈ D (A∗1) :

〈A∗1A∗2z, x〉 =Def. of A∗1

〈A∗2z,A1x〉 =Def. of A∗2

〈z,A2A1x〉 .

2. It remains to prove, that D((A2A1)

∗) ⊆ D (A∗1A∗2).

Let z ∈ D((A2A1)

∗). Since A2 ∈ L (Y,Z), for all x ∈ D (A2A1) = D (A1) :⟨(A2A1)

∗z, x⟩

=Def. of (A2A1)∗

〈z,A2A1x〉 =A2∈L(Y,Z)

〈A∗2z,A1x〉 .

⇒ A∗2z ∈ D (A∗1), i.e. z ∈ D (A∗1A∗2).

3. (a) D (A2A1) = A−11 D (A2)

(b) R(A−1

1

)= D (A1).

Since D (A2) ,D (A1) are dense, D (A2A1) is dense, ∀x ∈ D((A2A1)

∗), y ∈ D (A2A1) :

〈x,A2A1y〉 =⟨(A2A1)

∗x,A−1

1 A1y⟩

=⟨(A−1

1

)∗(A2A1)

∗x,A1y

⟩.

⇒

(a) x ∈ D (A∗2)

(b) A∗2x =(A−1

1

)∗(A2A1)

∗x =

by 2.(A∗1)

−1(A2A1)

∗x ∈ D (A∗1) .

⇒ x ∈ D (A∗1, A∗2) .

Theorem 2.7. Let Aj : D (Aj)→ Y,D (Aj) ⊆ X dense for X,Y Hilbert spaces, j ∈ 1, 2.

1. If A1 +A2 is densely defined on D (A1) ∩ D (A2), then (A1 +A2)∗ ⊇ A∗1 +A∗2.

2. If A2 ∈ L (X,Y ), then (A1 +A2)∗

= A∗1 +A∗2.

Proof. Exercise!

Remark. In general there is no equality in 1.

Definition 2.8. For vector spaces X,Y and a linear map A : D (A)→ Y,D (A) ⊆ X, we call

G (A) := (x,Ax) | x ∈ D (A)

the graph of A (c.f. FA I).

Remark. 1. If X,Y are normed spaces, then

‖(x, y)‖X×Y :=

√‖x‖2X + ‖y‖2Y or ‖x‖X + ‖y‖Y

define equivalent norms on X × Y .

17


2. If X,Y are Hilbert spaces, then

〈(x, y) , (x′, y′)〉 := 〈x, x′〉X + 〈y, y′〉Yis a scalar product on X × Y , which renders

(X × Y, ‖·‖X×Y

)a Hilbert space. The spaces

X ∼= X × 0 and Y ∼= 0 × Y are orthogonal subspaces in X × Y .

Definition 2.9. The norm ‖x‖A := ‖(x,Ax)‖X×Y is called the graph norm of x ∈ D (A).

Lemma 2.10. A subset G of X × Y is a graph of an operator A : D (A) → Y,D (A) ⊆ X, iff Gis a subspace with the property

(0, y) ∈ G ⇒ y = 0. (2.2)

Proof. “⇒” Evident.

“⇐” Suppose (x, y) , (x, y′) ∈ G. G sub-space=======⇒ (0, y − y′) ∈ G (2.2)

===⇒ y = y′.Hence for every x ∈ X there is at most one y ∈ X : (x, y) ∈ G. The map A : D (A)→ Ywith

D (A) := x ∈ X | ∃y ∈ Y : (x, y) ∈ G , Ax = y (2.3)is well defined and linear. By construction: G (A) = G.

Remark. The proof shows that there is a 1-to-1 correspondence between operators and graphs(with (2.2)).

2.1.2 Description of the adjoint through its graph

Consider the (obviously) isometric isomorphism

U : X × Y → Y ×X, U (x, y) := (y,−x) .

Theorem 2.11. Let A : D (A) → Y,B : D (B) → Y with D (A) ,D (B) ⊆ X dense on Hilbertspaces X,Y . Then

1. G (A∗) = U(G (A)

⊥)

= (UG (A))⊥.

2. B ⊆ A ⇒ A∗ ⊆ B∗.Proof. 1. By definition of A∗,

G (A∗) = (y, z) ∈ Y ×X | 〈y,Ax〉 = 〈z, x〉 ∀x ∈ D (A)=

(y, z) ∈ Y ×X | 〈(y, z) , (Ax,−x)〉Y×X = 0 ∀x ∈ D (A)

=

(y, z) ∈ Y ×X | 〈(y, z) ,U (v, w)〉Y×X = 0 ∀ (v, w) ∈ G (A)

= (UG (A))⊥

= U(G (A)

⊥).

2. We have

G (B) ⊆ G (A) ⇒ UG (B) ⊆ UG (A)

⇒ (UG (B))⊥⊇ (UG (A))

⊥

⇒ G (B∗) ⊇ G (A∗) .

There is a similar characterisation of the inverse using

V : X × Y → Y ×X, V (x, y) := (y, x) .

Theorem 2.12. Let A : D (A)→ Y,D (A) ⊆ X densely defined and injective.

1. For A−1 : R (A)→ X one hasG(A−1

)= VG (A) .

2. If R (A) ⊆ Y is dense, then A∗ : D (A∗)→ X is injective and (A∗)−1

=(A−1

)∗.Proof. 1. Evident.

2. Exercise.

18

2 Unbounded linear operators 2.2 Closed and closable operators

2.2 Closed and closable operators2.2.1 Definitions and basic results

Definition 2.13. For Banach spaces X,Y an operator A : D (A)→ Y,D (A) ⊆ X is called closed,iff its graph

G (A) := (x,Ax) | x ∈ D (A) ⊆ X × Y

is closed wrt. the product topology on X × Y (cf. FA I).

Recall. The product topology on X × Y is induced by the norm

‖(x, y)‖X×Y :=

√‖x‖2X + ‖y‖2Y

or any equivalent norm, e.g.

‖x‖X + ‖y‖y or max ‖x‖X , ‖y‖Y .

Proposition 2.14. The following are equivalent:

1. A is closed.

2. (D (A) , ‖·‖A) is complete.

3. If xnn ⊆ D (A) , xn → x,Axn → y, then x ∈ D (A) and Ax = y.

Proof. Evident.

Definition 2.15. An operator A : D (A) → Y,D (A) ⊆ X is called closable, iff the closure of thegraph G (A) (in X × Y ) is a graph of an operator. We denote this operator by A and call it theclosure of A.

Remark. Evidently, the following are equivalent:

1. A closable.

2. If xnn ⊆ D (A) is a null sequence and Axnn is convergent, then Axn → 0.

The closure A is uniquely determined by

D(A)

= x ∈ X | ∃ xnn ⊆ D (A) : xn → x, Axnn convergent ,Ax = lim

n→∞Axn.

For the relation of closed operators with bounded operators recall from FA I:

Theorem 2.16 (Closed Graph theorem – slight reformulation). Let A : D (A)→ Y,D (A) ⊆ X bea closed operator on Banach spaces X,Y . If D (A) is closed in X, then A is bounded.

Corollary 2.17. The following are equivalent:

1. A is closed and D (A) is closed in X.

2. A is closed and bounded.

3. A is bounded and D (A) is closed in X.

Proof. 1.⇒ 2. See closed graph theorem 2.16.

2.⇒ 3. Consider xnn ⊆ D (A) with xn → x. Then Axn → Ax, since A bounded and hencex ∈ D (A).

3.⇒ 1. Consider xnn ⊆ D (A) with xn → x and Axn → y. then x ∈ D (A), since D (A) isclosed and Axn → Ax, since A is bounded. Hence y = Ax.

19


2.2.2 Stability of closedness

Definition 2.18. Let A : D (A) → Y,B : D (B) → Y,D (A) ,D (B) ⊆ X be linear operators onBanach spaces X,Y . Then B is called A-bounded, iff

1. D (A) ⊆ D (B).

2. ∃a, b > 0 : ∀x ∈ D (A) :Bx ≤ a ‖Ax‖+ b ‖x‖ .

The infimum over all such b is called the A-bound of B.

Theorem 2.19. Let A : D (A) → Y,B : D (B) → Y,D (A) ,D (B) ⊆ X be linear operators onBanach spaces with B A-bounded with A-bound < 1. Then

A+B is closed ⇔ A is closed.

Proof. We will show that the graph norms ‖·‖A and ‖·‖A+B are equivalent on D (A) = D (A+B).For x ∈ D (A) :

‖Ax‖ ≤ ‖(A+B)x‖+ ‖Bx‖ (4-inequality)≤ ‖(A+B)x‖+ a ‖Ax‖+ b ‖x‖ (A-bound)

⇒ ‖x‖A = ‖x‖+ ‖Ax‖

≤ b

1− a‖x‖+

1

1− a‖(A+B)x‖

≤ max b, 11− a

‖x‖A+B

‖x‖A+B = ‖x‖+ ‖(A+B)x‖≤ ‖x‖+ ‖Ax‖+ ‖Bx‖ (4-inequality)≤ (1 + b) ‖x‖+ (1 + a) ‖Ax‖ (A-bound)≤ (1 + max a, b) ‖x‖A .

Hence (D (A) , ‖·‖A) is complete, iff(D (A) , ‖·‖A+B

)=(D (A+B) , ‖·‖A+B

)is complete.

In the following, we address the relation of closedness and the adjoint.

Theorem 2.20. Let A : D (A)→ Y,D (A) ⊆ X densely defined on Hilbert spaces X,Y . Then

1. A∗ is closed.

2. A is closable, iff D (A∗) is dense in Y .

In this case A = A∗∗.

3. If A is closable, then A∗

= A∗.

Proof. 1. Follows from G (A∗) = (UG (A))⊥.

2. Since

G (A) =(G (A)

⊥)⊥

=(U−1G (A∗)

)⊥= (x, y) ∈ X × Y | 〈x,A∗z〉 − 〈y, z〉 = 0 ∀z ∈ D (A∗) ,

we conclude:(0, y) ∈ G (A) ⇔ y ∈ D (A∗)

⊥.

20


By Lemma 2.10, this implies

G (A∗) is graph ⇔ D (A∗) ⊆ Y is dense.

In this case, by Theorem 2.11,

G (A∗∗) = U−1(G (A∗)

⊥)

= U−1U(G (A)

⊥⊥)

= G (A)⊥⊥

= G (A) = G(A).

3. Let A be closable. Again by Theorem 2.11,

G (A∗) = U(G (A)

⊥)

= U(G (A)

⊥)= U

(G(A)⊥)

= U(G(A))⊥

= G(A∗).

Example 2.21. Recall:

D0 : C∞c ((a, b))→ L2 (a, b) , D0f = f ′, (minimal differentiation operator)

Dmax : W 1,2 ((a, b))→ L2 (a, b) , Dmaxf = f ′. (maximal differentiation operator)

• Dmax is a closed operator in L2 (a, b), since D (Dmax) = W 1,2 ((a, b)) is complete wrt.

‖f‖W 1,2 =

√‖f‖22 + ‖f ′‖22 = ‖(f,Dmaxf)‖L2×L2 = ‖f‖Dmax

.

(cf. FA I).

• D0 is a closable operator in L2 ((a, b)), since

G (D0) = (f, f ′) | f ∈ C∞c ((a, b)) (FA I: C∞c ((a, b))‖·‖W1,2

= W 1,20 (a, b))

=

(f, f ′) | f ∈W 1,2 ((a, b)).

The graph’s closure G (D0) is the graph of the operator

D0 : W 1,20 ((a, b))→ L2 ((a, b)) .

Relation of their adjoints:

1. We have

D (D∗max) =

g ∈ L2 ((a, b)) | ∃g∗ ∈ L2 ((a, b)) :

ˆg∗f =

ˆgf ′ ∀f ∈W 1,2 ((a, b))

⊆

g ∈ L2 ((a, b)) | ∃c > 0 :

∣∣∣∣ˆ gf ′∣∣∣∣ ≤ c

√ˆ|f |2 ∀f ∈W 1,2 ((a, b))

= W 1,2 ((a, b)) . (cf. FA I)

Recall from FA I: ∀f, g ∈W 1,2 ((a, b)) :

ˆ b

a

gf ′ = g (b)f (b)− g (a)f (a)−ˆ b

a

g′f. (2.4)

HenceD (D∗max) =

g ∈W 1,2 ((a, b)) | g (a) = g (b) = 0

= W 1,2

0 ((a, b))

and thus,D∗maxg = −g′ ∀g ∈W 1,2

0 ((a, b)) , i.e. D∗max = −D0.

2. By (2.4), we have

D (D∗0) =

g ∈ L2 ((a, b)) | ∃g∗ ∈ L2 ((a, b)) :

ˆg∗f =

ˆgf ′ ∀f ∈ C∞c ((a, b))

⊇W 1,2 ((a, b))

and also ⊆W 1,2 ((a, b)), as W 1,2 is the maximal domain of the differentiation operator.Thus by (2.4),

D∗0g = −g′ ∀g ∈W 1,2 ((a, b)) , i.e. D∗0 = −Dmax.

Also, by Theorem 2.20, D0∗

= D∗0 = −Dmax.

21

2 Unbounded linear operators 2.3 Self-adjoint and normal operators

2.3 Self-adjoint and normal operatorsDefinition 2.22. A densely defined linear operator A : D (A)→ X,D (A) ⊆ X on a Hilbert spaceX is called

1. symmetric, iff A ⊆ A∗,

2. self-adjoint, iff A = A∗,

3. essentially self-adjoint, iff A∗ is self-adjoint.

Remark. A is essentially self-adjoint ⇒ A ⊆ A∗∗ = A∗ ⇒ A is symmetric.

Theorem 2.23. 1. Every symmetric operator A is closable with A ⊆ A∗.

2. The following are equivalent:

(a) A is essentially self-adjoint.

(b) A = A∗.

(c) A is self-adjoint.

Proof. 1. A∗ is a closed extension of A : G (A) ⊆ G (A∗) , hence G (A) is a graph and A ⊆ A∗.

2. By Theorem 2.20, A = A∗∗ and (as A is densely defined) A = A∗∗.

1.⇒ 2. A = A∗∗ = A∗.

2.⇒ 3. A = A∗∗2.= A

∗.

3.⇒ 1. A∗ = A∗ 3.

= A = A∗∗.

The criterion for boundedness of self-adjoint operators (which follows from the closed graphtheorem) was already discussed in FA I:

Theorem 2.24 (Hellinger-Tï¿œplitz). Every symmetric operator A defined on a Hilbert spaceD (A) = X is bounded.

For A ∈ L (X,Y ) on Hilbert spaces X,Y , A∗A is self-adjoint and positive. This remains truefor unbounded, closed A.

Theorem 2.25. Let A : D (A) → Y,D (A) ⊆ X be densely defined and closed on Hilbert spacesX,Y . then

A∗A : D (A∗A)→ X, D (A∗A) = x ∈ D (A) | Ax ∈ D (A∗)

is self-adjoint and positive. Moreover A|D(A∗A) = A.

Proof. For x ∈ D (A∗A) :

〈x, (1 +A∗A)x〉 = ‖x‖2 + ‖Ax‖2 ≥ ‖x‖2 ‖(1 +A∗A)x‖ ≥ ‖x‖ .

Hence, T := 1 +A∗A is injective. Since A is closed, we have

X × Y = G (A) + G (A)⊥

= G (A) + U−1G (A∗) .

∀z ∈ X : ∃x ∈ D (A) , y ∈ D (A∗) :

(z, 0) = (x,Ax) + U−1 (y,A∗y)

= (x,Ax) + (−A∗y, y)

= (x−A∗y,Ax+ y) .

Hence, z = x−A∗y and y = −Ax, i.e.

z = x+A∗Ax = Tx.

22

2 Unbounded linear operators 2.3 Self-adjoint and normal operators

In conclusion, T is surjective and thus bijective. ⇒ T−1 : X → X. Note that

T ∗ = 1 + (A∗A)∗ ⊇ 1 +A∗A∗∗ = 1 +A∗A = T,

as A is closed. By homework exercise,(T−1

)∗= (T ∗)

−1 ⊇ T−1.

Hence, T−1 : X → X is symmetric and bounded and as such self-adjoint. By homework exerciseA∗A = T − 1 is self-adjoint.

For A|D(A∗A) = A it suffices to show

G (A) = G(A|D(A∗A)

).

Let x ∈ D (A∗A). Then for g ∈ D (A∗A) :

〈x, Ty〉 = 〈x, (1 +A∗A) y〉 = 〈x, y〉+ 〈Ax,Ay〉 = 〈(x,Ax) , (y,Ay)〉 .

Since R (T ) = X, for all y ∈ D (A∗A) : (x,Ax) ⊥ (y,Ay) ⇒ x = 0.

Example 2.26. The operators −DmaxD0 and −D0Dmax, which both act as −f ′′, are self-adjointon their respective domain

D(−DmaxD0

)=f ∈W 1,2

0 ((a, b)) | f ′ ∈W 1,2 ((a, b)),

D(−D0Dmax

)=f ∈W 1,2 ((a, b)) | f ′ ∈W 1,2

0 ((a, b)),

are respectively called Dirichlet and Neumann Laplacian on L2 ((a, b)).

Remark. In case a = −∞ = −b, one has W 1,2 (R) = W 1,20 (R) (Meyers-Serin theorem) and hence

Dirichlet and Neumann Laplacian coincide.

Definition 2.27. A : D (A) → X,D (A) ⊆ X on a Hilbert space X is called normal iff D (A) =D (A∗) and

‖Ax‖ = ‖A∗x‖ ∀x ∈ D (A) .

Theorem 2.28. 1. Every normal operator is closed.

2. For a densely defined closed operator A, the following are equivalent:

(a) A is normal.

(b) A∗ is normal.

(c) AA∗ = A∗A.

Proof. 1. The graph norms of A and A∗ coincide on D (A) = D (A∗). Hence

(D (A) , ‖·‖A) = (D (A∗) , ‖·‖A∗) .

Since A∗ is closed, these spaces are complete, which by 2.14 implies that A is closed.

2. (a)⇔ (b) Evident.

(a)⇒ (c) For x ∈ D (A∗A) , y ∈ D (A) = D (A∗) :

〈A∗Ax, y〉 = 〈Ax,Ay〉 = 〈A∗x,A∗y〉 (by polarisation from ‖Ax‖ = ‖A∗x‖)

⇒ A∗x ∈ D (A∗∗) = D (A),

AA∗x = A∗∗A∗x = A∗Ax.

⇒ A∗A ⊆ AA∗. Similarly, AA∗ ⊆ A∗A.

23

2 Unbounded linear operators 2.4 Fundamentals of spectral theory

(c)⇒ (a) For x ∈ D (A∗A) = D (AA∗),

‖Ax‖2 = 〈Ax,Ax〉 = 〈x,A∗Ax〉 = 〈x,AA∗x〉 = 〈A∗x,A∗x〉 = ‖A∗x‖2 . (2.5)

According to Theorem 2.25,

A|D(A∗A) = A, A∗ |D(AA∗) = A∗.

Hence D (A) = D (A∗) and (2.5) holds for all x ∈ D (A).

Example 2.29 (Normal operators). 1. Every self-adjoint operator is normal.

2. Every unitary operator is normal.

3. Every maximal multiplication operator on L2 (Ω, µ) is normal.

Theorem 2.30. For every normal operator A:

1. A− z is normal for every z ∈ K. Moreover,

‖(A− z)x‖ = ‖(A∗ − z)x‖ ∀x ∈ D (A) = D (A∗) .

2. If A is injective, then A−1 is normal and A∗ is injective with (A∗)−1

=(A−1

)∗.3. R (A) = R (A∗).

Proof. 1. Exercise.

2. Since ‖Ax‖ = ‖A∗x‖ ∀x ∈ D (A) = D (A∗), N (A∗) = N (A) = 0 and A∗ is also injective.

Since R (A) = N (A∗)⊥

= X, A−1 : R (A) → X is densely defined and hence(A−1

)∗=

(A∗)−1 by homework exercise. Hence(A−1

)∗A−1 = (A∗)

−1A−1 = (AA∗)

−1= (A∗A)

−1= A−1 (A∗)

−1= A−1

(A−1

)∗,

i.e. A−1 is normal.

3. SinceR (A) = N (A)

⊥= N (A∗)

⊥= R (A∗),

the operators A and A∗ when restricted to R (A) are injective and adjoint.

Since the range is unchanged under this restriction, we may assume wlog. A injective.

By 2.,R (A) = D

(A−1

)= D

((A∗)

−1)

= R (A∗) .

2.4 Fundamentals of spectral theoryQuestion. For equations of the form

(A− z)x = y (2.6)

with A : D (A)→ X,D (A) ⊆ X on a Banach or Hilbert space X and z ∈ K.

1. Existence of solutions of (2.6) for every y ∈ X, i.e. (A− z) surjective?

2. Uniqueness of solutions of (2.6) for every y ∈ X, i.e. (A− z) injective?

3. Continuous dependence of solutions on y, i.e (A− z)−1 is continuous?

4. Continuous dependence of solutions on z, i.e. z 7→ (A− z)−1 is continuous on a domain ofzs for which (2.6) is uniquely solvable.

24


2.4.1 Spectrum and resolvent

Definition 2.31. For a linear operator A : D (A)→ X,D (A) ⊆ X on a normed space X, we call

ρ (A) :=z ∈ K | A− z : D (A→ X) is bijective and (A− z)−1is continuous

the resolvent set of A.

Note. If A is not closed, then ρ (A) = ∅, since either

1. A− z is not bijective, or

2. A − z is bijective s.t. (A− z)−1: X → X exists, but is not closed. But this implies that

(A− z)−1 is not continuous, since ∃ xnn ⊆ X :

xn → x, (A− z)−1xn → y,

but (A− z)−1x 6= y.

If X is a Banach space and A is closed, then by closed graph theorem 2.16:

ρ (A) = z ∈ K | A− z : D (A)→ X is bijective .

Definition 2.32. The resolvent function of A is

R (A, ·) : ρ (A)→ L (X) , z 7→ (A− z)−1.

The operator (A− z)−1 for z ∈ ρ (A) is called the resolvent of A at z. The spectrum of A is

σ (A) := K\ρ (A) .

Remark. The spectrum contains the set of eigenvalues z ∈ K of A, i.e. those z ∈ K for which A−zis not injective, i.e. N (A− z) 6= 0. Every x ∈ N (A− z) \ 0 is called eigenvector of A witheigenvalue z.

Lemma 2.33. For every densely defined closed operator A on a Hilbert space:

ρ (A∗) = ρ (A)∗, σ (A∗) = σ (A)

∗,

where M∗ := z | z ∈M .

Proof. For z ∈ ρ (A), the operator A − z is bijective. Hence A∗ − z is injective by homeworkexercise and

(A∗ − z)−1=[(A− z)−1

]∗∈ L (X) .

Hence A∗ − z is bijective, i.e. z ∈ ρ (A∗). This proves ρ (A) ⊆ ρ (A∗)∗. By A∗∗ = A, this also

proves ρ (A∗) ⊆ ρ (A)∗. The second identity σ (A∗) = σ (A)

∗ follows immediately.

2.4.2 Properties of the resolvent

Theorem 2.34 (Resolvent equations). Let A,B be linear operators on normed spaces.

1. For z, z′ ∈ ρ (A) :

R (A, z)−R (A, z′) = (z − z′)R (A, z)R (A, z′)

= (z − z′)R (A, z′)R (A, z) .(1stresolvent equation)

2. For z ∈ ρ (A) ∩ ρ (B) :

R (A, z)−R (B, z) =

R (A, z) (B −A)R (B, z) , if D (B) ⊆ D (A)

R (B, z) (B −A)R (A, z) , if D (A) ⊆ D (B).

(2ndresolvent equation)

25


Proof. It suffices to prove the 2ndresolvent equation in D (B) ⊆ D (A). Wlog. z = 0 ∈ ρ (A)∩ρ (B).

A−1 = A−1BB−1 = A−1 [B −A+A]B−1

= A−1 [B −A]B−1 +A−1AB−1 (R(B−1

)= D (B) ⊆ D (A))

= A−1 (B −A)B−1 +B−1.

Theorem 2.35 (Stability of invertibility). For linear operators A : D (A) → Y,B : D (B) → Y ,D (A) ⊆ D (B) ⊆ X on Banach spaces X,Y , if A is closed and bijective and

∥∥BA−1∥∥ < 1, then

A+B is closed and bijective with

(A+B)−1

=

∞∑n=0

(−1)nA−1

(BA−1

)n=

∞∑n=01

(−1)n (A−1B

)nA−1,

where the series converges in norm in L (Y,X).

Proof. B is A-bounded with A-bound < 1, since ∀x ∈ D (A) :

‖Bx‖ ≤∥∥BA−1Ax

∥∥ ≤ <1︷︸︸︷∥∥BA−1∥∥ ‖Ax‖ < ‖Ax‖ .

Hence A+B is closed by 2.19. For any 0 6= x ∈ D (A+B) = D (A) :

‖(A+B)x‖ ≥ ‖Ax‖ − ‖Bx‖ ≥(1−

∥∥BA1−∥∥) ‖Ax‖ > 0.

Hence A+B is injective. We will show that the sequence

TN :=

N∑n=0

(−1)nA−1

(BA−1

)nconverges in norm to some T ∈ L (Y,X) with

(A+B)T = 1.

In particular this implies that (A+B)is surjective with (A+B)−1

= T . For N > M :

‖TN − TM‖ =

∥∥∥∥∥N∑

n=M+1

(−1)nA−1

(BA−1

)n∥∥∥∥∥≤

N∑n=M+1

∥∥A−1∥∥∥∥BA−1

∥∥n=∥∥A−1

∥∥ ∥∥BA−1∥∥M+1

N−M−1∑n=0

∥∥BA−1∥∥n

=∥∥A−1

∥∥ ∥∥BA−1∥∥M+1 1

1− ‖BA−1‖−−−−→M→∞

0.

Hence TN is Cauchy in L (Y,X), i.e. ∃T ∈ L (Y,X) : ‖TN − T‖ → 0. Moreover,

(A+B)TN =

N∑n=0

(−1)n

(A+B)A−1(BA−1

)n= 1 +

N∑n=1

(−1)n (BA−1

)n+

N+1∑n=1

(−1)n−1 (

BA−1)n

= 1 + (−1)N (

BA−1)N+1 −−−−→

N→∞1.

Hence for all y ∈ Y :TNy → Ty and (A+B)TNy → y.

Since A+B is closed, this implies that Ty ∈ D (A+B) and (A+B)Ty = y.

26

2 Unbounded linear operators 2.5 Spectra of symmetric and self-adjoint operators

Theorem 2.36. Let A : D (A)→ X,D (A) ⊆ X be a closed linear operator on a Banach space X.

1. ρ (A) is an open and σ (A) a closed subset of K.

2. For any z ∈ ρ (A) : d (z, σ (A)) ≥ 1‖R(A,z)‖ .

3. If z0 ∈ ρ (A) and |z − z0| < 1‖R(A,z0)‖ , then z ∈ ρ (A) and

R (A, z) =

∞∑n=0

(z − z0)nR (A, z0)

n+1.

Note. By 3., the resolvent is analytic in ρ (A)!

Proof. 1. For z0 ∈ ρ (A) , r := 1‖R(A,z0)‖ and |z − z0| < r,

‖(z − z0)R (A, z0)‖ < 1.

Hence A− z = A− z0 + z − z0 is bijective by 2.35 with B = z − z0, s.t. z ∈ ρ (A).

2. From 1., |z − z0| > r for any z ∈ σ (A) = K\ρ (A).

3. By by 2.35, for |z − z0| < r :

R (A, z) =

∞∑n=0

(−1)nR (A, z0) [(z0 − z)R (A, z0)]

n

=

∞∑n=0

(z − z0)nR (A, z0)

n+1.

2.5 Spectra of symmetric and self-adjoint operatorsFamiliar from Linear Algebra.

Lemma 2.37. The eigenvalues of symmetric operators are real. For symmetric operators andnormal operators, the eigenvectors corresponding to distinct eigenvalues are orthogonal.

Proof. If Ax = λx for some x ∈ D (A) \ 0, then

λ ‖x‖2 = 〈x, λx〉 = 〈x,Ax〉 = 〈Ax, x〉 = 〈λx, x〉 = λ ‖x‖2

⇒ λ ∈ R.If Ax = λx and Ay = µy for x, y ∈ D (A) \ 0 with λ 6= µ, then

1. if A symmetric:

(λ− µ) 〈x, y〉 = 〈λx, y〉 − 〈x, µy〉 = 〈Ax, y〉 − 〈x,Ay〉 = 0.

2. if A normal:(λ− µ) 〈x, y〉 =

⟨λx, y

⟩− 〈x, µy〉 = 〈A∗x, y〉 − 〈x,Ay〉 = 0.

Lemma 2.38. Let A be a symmetric operator on a complex Hilbert space.

1. If z ∈ C\R, then A− z is continuously invertible with

‖(A− z)x‖ ≥ |Im z| ‖x‖ ∀x ∈ D (A) and∥∥∥(A− z)−1

∥∥∥ ≤ 1

|Im z|.

2. If A is closed, then R (A, z) is closed for every z ∈ C\R.

27


Proof. 1. ∀x ∈ D (A) :

‖(A− z)x‖ ‖x‖ ≥ |〈x, (A− z)x〉| (Cauchy-Schwarz)

=∣∣∣〈x, (A− Re z)x〉︸︷︷︸

∈R

− i Im z ‖x‖2∣∣∣

≥ |Im z| ‖x‖2 .

This implies that (A− z) is injective and ∀x ∈ R (A− z) :∥∥∥(A− z)−1x∥∥∥ ≤ 1

|Im z|‖x‖ , i.e.

∥∥∥(A− z)−1∥∥∥ ≤ 1

|Im z|.

2. (A− z)−1 is closed and continuous by 1. Hence R (A− z) = D(

(A− z)−1)is closed.

Remark. A− z is in general not bijective.

The following is an important criterion on symmetric operators to be self-adjoint or essentiallyself-adjoint.

Theorem 2.39 (Criterion for (essential) self-ajointness). Let A be a symmetric operator on acomplex Hilbert space X.


(a) A is self-adjoint.

(b) R (A− z) = X for all z ∈ C\R.(c) ∃z± ∈ C with Im z+ > 0 and Im z− < 0 : R (A− z±) = X.

(d) A is closed and ∃z± ∈ C, Im z+ > 0, Im z− < 0 : N (A∗ − z±) = 0.(e) σ (A) ⊆ R.


(a) A is essentially self-adjoint.

(b) R (A− z) = X for all z ∈ C\R.(c) ∃z± ∈ C, Im z+ > 0, Im z− < 0 : R (A− z±) = X.

(d) ∃z± ∈ C, Im z+ > 0, Im z− < 0 : N (A∗ − z±) = 0.(e) σ

(A)⊆ R.

Proof. 1. (a)⇒ (b) If R (A− z) 6= X, then

N (A− z) = N (A∗ − z) = R (A− z)⊥ 6= 0 ,

since R (A− z) is closed. This is a contradiction to the continuous invertibilityof A− z (cf. Lemma 2.38).

(b)⇒ (a) Since A ⊆ A∗, it suffices to show D (A∗) ⊆ D (A).

• Pick z ∈ C\R. Then A− z and A− z are bijective (i.e. injective by Lemma2.38 and surjective by assumption). Hence D

((A− z)−1

)= X.

• Pick x ∈ D (A∗). Then x0 := (A− z)−1(A∗ − z)x ∈ D (A) and

Ax0 = A∗x0, (A∗ − z)x = (A− z)x0

⇒ (A∗ − z) (x− x0) = (A∗ − z)x− (A− z)x0 = 0

i.e. x− x0 ∈ N (A∗ − z) = R (A− z)⊥ = 0⇒ x = x0 ∈ D (A) .

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(b)⇒(c) Evident.

(a)⇒ (d) SinceA is self-adjoint, it is closed. Moreover, since (a)⇒ (c) we haveN (A∗ − z) =

R (A− z)⊥ = 0.

(d)⇒ (c) Since A is closed R (A− z) = R (A− z) = N (A∗ − z)⊥ = X.

(c)⇒ (e) Since A−z± is injective andR (A− z±) = X by Lemma 2.38, we have z± ∈ ρ (A).By Theorem 2.36, for every |z − z±| < Im z± ≤ 1

‖(A−z±)−1‖ , we have z ∈ ρ (A).

Repeating the argument allows to reach every point in C\R s.t. C\R ⊆ ρ (A).

(e)⇒ (b) Evident.

2. Homework.

Applications

1. Let (X,µ) be a σ-finite measure space and consider the maximal multiplication operatorMtf = tf with

D (Mt) =f ∈ L2 (X,µ) | tf ∈ L2 (X,µ)

for t : X → R measurable. Obviously Mt is symmetric.

One proof of self-adjointness. Let h = t± i and for f ∈ L2 (X,µ), let

f1 :=f

h. (well defined, as t ∈ R)

Then f1 ∈ D (Mh) = D (Mt) and f = Mhf1 ∈ R (Mt ± i). Hence

R (Mt ± i) = L2 (X,µ) .

2. ConsiderP : C∞c (R)→ L2 (R) , Pf := −if ′.

Then P is densely defined and symmetric, as for f, g ∈ C∞c (R) :

〈g, Pf〉 =

ˆg (−if ′) = −igf |∞−∞︸︷︷︸

=0

+i

ˆg′f = 〈Pg, f〉 .

P is not self-adjoint, since e.g. f (x) = e−x2

is in D (P ∗), but not in D (P ) = C∞c (R).

However, P is essentially self-adjoint since for g ∈ N (P ∗ ± i) ⊆ D (P ∗) ⊆W 1,2 (R), i.e.ˆ

(−ig′ ± ig) f = 0 ∀f ∈ C∞c (R) ,

the fundamental lemma of variational calculus (FA I) yields

−ig′ ± ig = 0 or g′ = ±g, i.e. g (x) = C · e±x,

which implies, g /∈ L2 (R) unless C = 0.

Theorem 2.40 (Characterisation of the spectrum). Let A be self-adjoint or normal in a (real orcomplex) Hilbert space.

1. z ∈ K is an eigenvalue, iff R (A− z) 6= X.

The corresponding eigenspace is N (A− z) = R (A− z)⊥.

2. The following are equivalent for z ∈ K.

(a) z ∈ σ (A).

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2 Unbounded linear operators 2.6 Point, continuous and residual spectrum

(b) ∃ xnn ⊆ D (A) with ‖xn‖ = 1 and (A− z)xn −−−−→n→∞

0.

(c) R (A− z) 6= X.

Proof. Since every self-adjoint operator is normal, we establish these properties for normal oper-ators only. Recall by Theorem 2.30 for normal A :

‖(A− z)x‖ = ‖(A∗ − z)x‖ ∀x ∈ D (A) = D (A∗)

N (A− z) = N (A∗ − z) ,

R (A− z) = R (A∗ − z).

1. “⇒” If z is eigenvector of A, then

N (A− z) = N (A∗ − z) 6= 0

and henceR (A− z) = N (A∗ − z)⊥ 6= X.

“⇐” If R (A− z) 6= X, then

N (A− z) = N (A∗ − z) = R (A− z)⊥ 6= 0 .

2. (a)⇒ (c) If R (A− z) = X,then

N (A− z) = N (A∗ − z) = R (A− z)⊥ = 0

and hence A− z is injective with

D(

(A− z)−1)

= R (A− z) = X, s.t. (A− z)−1 ∈ L (X) .

This implies z ∈ ρ (A).(c)⇒ (b) In case R (A− z) = X, then A− z is injective (see 1.) and

D(

(A− z)−1)

= R (A− z)

is not closed. In particular, A− z is not continuously invertible.Hence, ∃ ynn ⊆ R (A− z) and y /∈ R (A− z) , ‖y‖ = 1 s.t.

yn → y and∥∥∥(A− z)−1

yn

∥∥∥→∞.The sequence xn := (A−z)−1yn

‖(A−z)−1yn‖ satisfies the requirement.

In case R (A− z) 6= X, A− z is not injective and we pick

xn ∈ N (A− z) , ‖xn‖ = 1,

which satisfy the requirement.(b)⇒ (a) Homework (evident).

2.6 Point, continuous and residual spectrumRecall for closed, densely defined operator A on a Banach space X,

ρ (A) := z ∈ K | A− z bijective .

Definition 2.41. For the above setting

1. σp (A) := z ∈ K | A− z is not injective is called the point spectrum of A,

2. σr (A) :=z ∈ K | A− z injective, R (A− z) 6= X

is called the residual spectrum of A,

3. σc := σ (A) \ (σp (A) ∪ σr (A)) is called continuous spectrum of A.

Note. σ (A) = σp (A)∪ σr (A)∪ σc (A) is a disjoint union. Moreover, σp (A) is the union of the setof eigenvalues.

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2 Unbounded linear operators 2.6 Point, continuous and residual spectrum

WARNING! For some applications, in particular in mathematical physics, one also defines forself-adjoint operators A:

σpp (A) := σp (A), (pure point spectrum)σc (A) := σac (A) ∪ σsc (A) , (continuous spectrum)

where σac (A) and σsc (A) are the absolutely continuous and singular continuous spectra (morelater...).

Corollary 2.42. For any self-adjoint or normal operator A in a Hilbert space, σr (A) = ∅.

Proof. Since N (A− z) = N (A∗ − z) = 0, if A− z is injective,

R (A− z) = N (A∗ − z)⊥ = X.

Example 2.43 (Spectrum of the maximal multiplication operator).

Claim. Mt − z is injective, iff µ (x ∈ X | t (x) = z) = 0.

Proof. In case t (x) 6= z for µ-almost every x, then

Mt−zf (x) = (t (x)− z) f (x) 6= 0 ∀f 6≡ 0.

In case µ (x ∈ X | t (x) = z) > 0, then

Mt−zf = 0 ∀f ∈ L2 (X,µ) with supp f ⊆ t (x) = z ,

i.e. Mt−z is not injective.

That is: z is an eigenvalue of Mt, iff µ (x ∈ X | t (x) = z) > 0.

Claim. z ∈ σ (Mt), iff ∀ε > 0 :

µ (x ∈ X | |t (x)− z| < ε) > 0.

Proof. Exercise.

E.g.Mxf (x) = xf (x) , D (Mx) =

f ∈ L2 ((0, 1)) | xf ∈ L2 ((0, 1))

has only continuous spectrum σ (Mx) = σc (Mx) = [0, 1].

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Spectral Theory in Hilbert Spaces [MA5092]1em SS 2013 · Spectral Theory in Hilbert Spaces [MA5092]...

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