8 Operators on Hilbert spaces
8.1 Algebra
We will start with a little bit of algebra recollection:
Definition 8.1. Given a field F an (associative F-) algebra is a vector space A over F equipped with a bilinear operation · : A × A → A, such that (A, ·) is a monoid, with a neutral element denoted by 1. A Banach algebra is an algebra A which is also a Banach space and the norm satisfies
kx · yk ≤ kxk kyk ∀ x, y ∈ A.
A ∗-algebra is a C-algebra A, equipped with a map ∗ : A → A with the following properties:
i) 1∗ = 1;
ii) x∗∗ = x for all x ∈ A;
iii) (x · y)∗ = y∗ · x∗ for all x, y ∈ A;
iv) conjugate-linearity, i.e. (ax + by)∗ = ax∗ + by∗ for all x, y ∈ A, a, b ∈ C.
The map ∗ : A → A is called an involution. A Banach algebra, which is also a ∗-algebra A is called a C∗-algebra if the norm satisfies
kx∗xk = kxkkx∗k x ∈ A.
Definition 8.2. A subspace I ⊆ A of an algebra A is is called:
i)a left ideal, if A · I ⊆ I;
ii)a right ideal, if I · A ⊆ I;
iii)a two-sided ideal, if it is both a left and a right ideal.
A ∗-ideal I in a ∗-algebra A is an ideal, such that I∗ ⊆ I.
8.2 Hilbert spaces
Definition 8.3. A Hilbert space H is a vector space over a field F ∈ {R, C} equipped with a inner product h·, ·i → F, characterized by
1. linearity in the first argument;
2. conjugate symmetry, i.e hx, yi = hy, xi;
1 3. positivity, i.e. hx, xi > 0 for all x ∈ H.
Moreover, H is a complete metric space with respect to the norm defined by the inner product, i.e. every Cauchy sequence in H has a limit in H.
From now we will consider only separable, infinite dimensional, complex Hilbert spaces.
Examples:
1. We denote n o P∞ 2 `2 := {xn}n∈N xn ∈ C, n=1 |xn| < +∞ .
The set `2 can be equipped with a Hilbert space structure with the product
∞ X hx, yi := xnyn. n=1
2. Let (X, µ) be a measurable space. We denote
2 n R 2 o L (X, µ) := f : X → C f is measurable and X |f(x)| dµ < +∞ .
The set L2(X, µ) can be equipped with a Hilbert space structure with the product Z hf, gi := f(x)g(x)dµ. X
3. Let (X, µ) be a measurable space and let H be a Hilbert space. Denote
2 n R 2 o L (X, µ, H) := f : X → H f is measurable and X kf(x)kHdµ < +∞ .
The set L2(X, µ) can be equipped with a Hilbert space structure with the product Z hf, gi := hf(x), g(x)iHdµ. X
Exercise 8.1. Prove that the inner product on a complex Hilbert space can be recovered from the norm by the polarization identity: 1 hx, yi = kx + yk2 − kx − yk2 − ikx + iyk2 + ikx − iyk2 ∀ x, y ∈ H. 4
Definition 8.4. We say that two Hilbert spaces H1 and H2 are isomorphic if there exists a vector space isomorphism A : H1 → H2, such that
hA · ,A · iH2 = h·, ·iH1 .
In that case A is called an unitary operator.
2 Lemma 8.5 (Riesz Lemma). Denote by H∗ the space of linear, continuous, complex-valued ∗ functionals on H. Then for every T ∈ H there exists xT ∈ H, such that for all y ∈ H we have:
T (y) = hy, xT i.
Definition 8.6. We say that a set S ⊆ H is orhonormal if for all x ∈ S, kxk = 1 and for all x, y ∈ S, hx, yi = 0. An orthonormal subset S ⊆ H is called an orhonormal basis, if
x ∈ H hx, yi = 0 ∀ y ∈ S = {0}.
Theorem 8.7. A Hilbert space is separable if and only if it has a countable orthonormal basis. If the basis S is finite, then the Hilbert space is isomorphic to Cn for n := #S. If the basis is infinite, then the Hilbert space is isomorphic to `2.
8.3 Operators on Hilbert spaces
From now on, we will consider only linear operators on the Hilbert space H with dense domain, i.e. A : D(A) → H, such that D(A) = H.
Definition 8.8. Let A be a linear operators on H with dense domain. We define the domain of the adjoint operator by n o ∗ D(A ) := x ∈ H ∃ yx ∈ H hAz, xi = hz, yxi ∀ z ∈ D(A) .
∗ ∗ ∗ Then we can define the adjoint operator A : D(A ) → H by setting A (x) = yx.
Definition 8.9. We say that a linear oparator A : D(A) → H is:
Closed if its graph Γ(A) := {(x, Ax) | x ∈ D(A)} is a closed subset of H;
Bounded if there exists c ∈ R, such that kAxk ≤ ckxk for all x ∈ D(A). For a bounded operator A we denote its norm by
n kAxk o kAk := sup kxk x ∈ D(A), x 6= 0 . (8.1)
We denote the set of all closed and bounded operators on H by B(H).
Symmetric if for all x, y ∈ D(A), we have hAx, yi = hx, Ayi;
Self-adjoint if A = A∗ or, equivalently, A is symmetric and D(A∗) = D(A). We denote the set of
self-adjoined linear operators by Lad(H) and Bad(H) := B(H) ∩ Lad(H);
Normal if AA∗ = A∗A;
∗ Isometric if A A = IdD(A);
3 ∗ ∗ Unitary if A A = IdD(A) and AA = IdD(A∗);
Positive if hAx, xi ≥ 0 for all x ∈ D(A). We denote it by A ≥ 0;
Compact if A ∈ B(H) and A maps bounded sets into pre-compact sets. We denote the set of
compact operators by B∞(H);
Finite rank if the image of A is finite dimensional;
A projection if A ∈ B(H) and A2 = A;
An orthogonal projection if it is a self-adjoint projection. We denote the set of orthogonal projections
by Bop(H).Naturally, Bop(H) ⊆ B(H).
Exercise 8.2. Show that A is a closed operator if and only if D(A) with the graph scalar product
hx, yiA = hx, yi + hT x, T yi, is a Hilbert space.
Definition 8.10. We say that an operator A is an extension of an operator B if D(B) ⊆ D(A) and Ax = Bx for all x ∈ D(B).
Exercise 8.3. For every densely defined operator A, A∗∗ is an extension of A.
Exercise 8.4. Show that A is a closed operator if and only if A∗∗ = A.
Exercise 8.5. Let A be a symmetric operator. Prove that the following are equivalent:
a) A∗∗ is self-adjoint;
b) Ker(A∗ + i Id) = Ker(A∗ − i Id) = {0};
c) Im(A∗ + i Id) = Im(A∗ − i Id) = H.
Theorem 8.11 (Closed graph theorem). Let A be a closed linear operator in a Hilbert space H, and D(A) = H. Then A is bounded.
Exercise 8.6. Show that a symmetric operator A with D(A) = H is bounded and self- adjoined.
Exercise 8.7. Let A be a closed, bounded linear operator, with dense domain in a Hilbert space H. Show that D(A) = H.
Exercise 8.8. Show that for every operator A ∈ B(H), A∗ exists, is closed and D(A∗) = H. Moreover, we have A∗∗ = A.
Exercise 8.9. Prove that B(H) with the operator norm k · k as in (8.1) is a Banach space.
4 Exercise 8.10. Prove that B(H) with the operator norm k · k as in (8.1) has a structure of a C∗-algebra the composition of operators as the product, the adjoint operation as the involution, and the identity operator Id as a unit.
Exercise 8.11. Show that B∞(H) is a two-sided ideal in B(H).
Theorem 8.12. A positive linear operator A satisfies the Cauchy-Bunyakowsky-Schwartz inequality: |hAx, yi|2 ≤ hAx, xihAy, yi ∀ x, y ∈ D(A).
Exercise 8.12. Show that every bounded, positive linear operator is self-adjoined. Hint: Use the polarization identity form Exercise 8.1.
Lemma 8.13 (Square root lemma). Let A ∈ B(H) and A ≥ 0. Then there exists a unique B ∈ B(H), such that B ≥ 0 and B2 = A. Moreover, B commutes with all operators from √ B(H) that A commutes with. We denote A := B.
Exercise 8.13. Show that if A ∈ B(H), then A∗A ≥ 0.
Definition 8.14. We define a map | · | : B(H) → B(H) by taking √ |A| := A∗A.
Exercise 8.14. Let A ∈ B(H). Prove that the following are equivalent:
i) A ∈ B∞(H);
∗ ii) A ∈ B∞(H);
∗ iii) A A ∈ B∞(H);
iv) |A| ∈ B∞(H);
v) A maps every weakly convergent sequence to a convergent sequence. In other words,
if limn→∞hxn, yi = hx, yi for all y ∈ H, then limn→∞ kAx − Axnk;
vi) there exists a sequence of finite rank operators An, such that limn→∞ kA − Ank = 0.
Exercise 8.15. Show that if An is a sequence of compact operators and limn→∞ kA−Ank = 0, then A is also compact.
Definition 8.15. The regular set of a closed operator A with dense domain is defined n o
ρ(A) := λ ∈ C A − λ Id : D(A) → H is a bijection .
−1 For λ ∈ ρ(A) the operator Rλ := (A − λ Id) is called the resolvent of A at λ. The complement σ(A) := C \ ρ(A) is called the spectrum of A. The set n o
σp(A) := λ ∈ C Ker(A − λ Id) 6= {0} ,
5 is called the point spectrum of A. An element of σp(A) is called an eigenvalue of A. For an eigenvalue λ and element of Ker(A−λ Id) is called an eigenvector of A with eigenvalue λ. For an eigenvalue λ the dimension Ker(A − λ Id) is called the multiplicity of λ.
In other words, n o
σp(A) := λ ∈ C A − λ Id : D(A) → H is not injective 6= {0} .
Naturally, σp(A) ⊆ σ(A).
Exercise 8.16. Show that for a closed operator A with dense domain its regular set ρ(A) is open in C.
8.4 Riesz-Schauder and Hilbert-Schmidt theorems
Theorem 8.16 (Riesz-Schauder Theorem). The spectrum of a compact operator A ∈
B∞(H) has the following properties:
i) is countable;
ii) has no accumulation points accept 0;
iii) 0 ∈ σ(A);
iv) σ(A) = σp(A);
v) for every λ ∈ σp(A) the multiplicity is finite;
∗ vi) if λ ∈ σp(A), then λ ∈ σp(A ) and
dim Ker(A − λ Id) = dim Ker(A∗ − λ Id).
Theorem 8.17 (Hilbert-Schmidt Theorem). Let A ∈ B∞(H) be a normal operator. Then there exists a complex sequence λn ∈ C and an orthonormal basis en ∈ H, such that limn→∞ λn = 0 and
∗ Aen = λnen and A en = λnen ∀ n ∈ N.
Exercise 8.17. Let {en}n∈N be an orthonormal basis of H and let {λn}n∈N be a bounded sequence in C. Define an operator A as the linear extension of Aen = λnen. Show that then A is normal. Moreover, A is compact if and only if limn→∞ λn = 0.
You can find more on Hilbert spaces in Volumes II and VI of Reed’s and Simon’s book Modern Methods of Mathematical Physics.
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