FROM THE DEFINITION OF A HILBERT SPACE TO THE PROOF OF SPECTRAL THEOREMS

Maciej Horowski Institute of Physics, Warsaw University Division 15-424 Bialystok, Lipowa 41, Poland

1 Hilbert spaces

1.1

Let H be a complex vector space equipped with a scalar product denoted by (·|·), so (H, (·|·)) is a unitary space. Every scalar product (·|·) defines a norm ∥ · ∥ by

∥x∥^{2} =^{df} (x|x) x ∈ H.

Definition 1.1.1 A complete unitary space is called a Hilbert space.

It will be usefull to recall the two following inequalities:

a) Schwarz inequality

∀x, y ∈ H |(x|y)|^{2} ≤ (x|x)(y|y);

b) Minkowski inequality

∀x, y ∈ H ∥x + y∥ ≤ ∥x∥ + ∥y∥.

1.2 Examples of Hilbert spaces

a) C^{n} with the standard scalar product:

(z|w) =^{df} ∑_{i=1}^{n} z̄_{i}w_{i} ∀z = (z_{1}, …, z_{n}), w = (w_{1}, …, w_{n}) ∈ C^{n};

b)

l^{2} =^{df} {{a_{n}} ⊂ C, n ∈ N : ∑_{n} |a_{n}|^{2} < ∞} ,

with the scalar product given by:

({a_{n}}|{b_{n}}) =^{df} ∑_{n} ā_{n}b_{n};

c) L^{2}(R^{n}, dλ), where dλ is the Lebesgue measure, with the standard scalar product:

(f|g) =^{df} ∫_{R^{n}} \overline{f(x)}g(x)dλ(x), f, g ∈ L^{2}(R^{n}, dλ).

1.3

Definition 1.3.1 a) A Hilbert space is called separable if there exists a countable set which is dense in this space or, equivalently, this space has a countable basis.

b) Two vectors x, y ∈ H are called orthogonal if (x|y) = 0; we denote it by x ⊥ y.

c) A set in a Hilbert space is called orthogonal if any two elements of this set are orthogonal.

d) If the norm of any element of an orthogonal set is one, this set is called orthonormal.

e) An orthonormal (o.n.) basis of a Hilbert space is any complete orthonormal set.

1.4

Proposition 1.4.1 If H_{1} and H_{2} are Hilbert spaces, we have a Hilbert space structure on H_{1} × H_{2} given by:

((x, y)|(x′, y′))_{H_{1}×H_{2}} =^{df} (x|x′)_{H_{1}} + (y|y′)_{H_{2}} ∀(x, y), (x′, y′) ∈ H_{1} × H_{2}.

□

Definition 1.4.1 The Hilbert space H_{1} × H_{2} with the scalar product defined above is called a direct sum of Hilbert spaces and we denote it by H_{1} ⊕ H_{2}.

Proposition 1.4.2 A scalar product as a function (·|·) : H × H → C is continuous.

\underline{Proof} If x_{n} → x and y_{n} → y, then {x_{n}}, as a convergent sequence is bounded.

|(x_{n}|y_{n}) − (x|y) − (x_{n}|y) + (x_{n}|y)| ≤
≤ |(x_{n}|y_{n} − y) + (x_{n} − x|y)| ≤
≤ M∥y_{n} − y∥ + ∥y∥∥x_{n} − x∥.
□

1.5

Proposition 1.5.1 If {e_{n}}_{n∈N} is an o.n. basis and {t_{n}} ∈ l^{2}, the series ∑_{n} t_{n}e_{n} is convergent and

∥ ∑_{n} t_{n}e_{n}∥ = (∑_{n} |t_{n}|^{2})^{\frac{1}{2}} .  (1)

\underline{Proof}

Let S_{N} =^{df} ∑_{n=1}^{N} t_{n}e_{n}, then

∥S_{N} − S_{M}∥^{2} = ∥ ∑_{n=M+1}^{N} t_{n}e_{n}∥^{2} =

= (t_{M+1}e_{M+1} + ... + t_{N}e_{N}|t_{M+1}e_{M+1} + ... + t_{N}e_{N}) = ∑_{n=M+1}^{N} |t_{n}|^{2}.

Since the series ∑_{n} |t_{n}|^{2} is convergent, the sequence {S_{N}} is Couchy, so it is convergent. Since the scalar product is continuous, we have

(∑_{n} t_{n}e_{n}| ∑_{m} t_{m}e_{m}) = ∑_{n,m} t̄_{n}t_{m}(e_{n}|e_{m}) = ∑_{n} |t_{n}|^{2}.

□

Proposition 1.5.2 If {e_{n}} is an o.n. basis, x ∈ H and t_{n} =^{df} (e_{n}|x), then

a) {t_{n}} ∈ l^{2};

b) x = ∑_{n} e_{n}t_{n};

c) ∥x∥^{2} = ∑_{n} |t_{n}|^{2}.

□

1.6 Example. Fourier series

Let us consider L^{2}([0, 2π]) with a scalar product given by

(f|g) =^{df} \frac{1}{2π} ∫_{0}^{2π} f̄(λ)g(λ)dλ.

The family of functions defined by

e_{k}(λ) =^{df} e^{iλk}, k ∈ Z,

form an orthonormal set and we shall show that it is an o.n. basis: Let as assume that there exists f ∈ L^{2}([0, 2π]) such that (f|f) = 1 and (e_{k}|f) = 0 ∀k ∈ Z. Then we have

0 = (e_{k}|f) = \frac{1}{2π} ∫_{0}^{2π} e^{−ikλ}f(λ)dλ ⇒

⇒ ∫_{0}^{2π} ∑_{k=−N}^{N} a_{k}e^{−ikλ}f(λ)dλ = 0 ∀N ∈ N, ∀a_{k} ∈ C.

Inasmuch as the family of functions of the form ∑_{k=−N}^{N} a_{k}e^{−iλk} fulfil the assumptions of the Stone– Weierstrass theorem on a circle, so by this theorem f(λ) ≡ 0, which is contradiction. □

Now let as consider the consequence of the Proposition 1.5.2:

f ∈ L^{2}([0, 2π]) ⇒ t_{k} = \frac{1}{2π} ∫_{0}^{2π} e^{−ikλ}f(λ)dλ,

moreover

\frac{1}{2π} ∫_{0}^{2π} |f(λ)|^{2}dλ = ∑_{k=−∞}^{∞} |t_{k}|^{2},

finally we obtain the Fourier series

f(λ) = ∑_{k=−∞}^{∞} t_{k}e^{ikλ},

which means that

\frac{1}{2π} ∫_{0}^{2π} |f(λ) − ∑_{k=−N}^{N} t_{k}e^{ikλ}|^{2}dλ ⟶ 0

if N → ∞.

2 Subspaces

2.1

Definition 2.1.1 A vector subspace F of a Hilbert space H is closed if F is closed in the topology generated by the scalar product in H.

2.2

Definition 2.2.1 Let G be any subset of H, then

G^{⊥} =^{df} {x ∈ H : ∀y ∈ G (x|y) = 0}.

Proposition 2.2.1 Let G be any subset of H, then

a) G^{⊥} is a closed subspace;

b) G ⊂ G^{⊥⊥}.

□

2.3

Theorem 2.3.1 (on the orthogonal projection (Beppo-Levi)) If F is a closed subspace of H and x ∈ H, then

a) x = x_{∥} + x_{⊥}, where x_{∥} ∈ F, x_{⊥} ∈ F^{⊥};

b) this decomposition is unique.

\underline{Proof}

a) Let {e_{n}} be an o.n. basis in F and let

x_{∥} =^{df} ∑_{n}(e_{n}|x)e_{n} and x_{⊥} =^{df} x − x_{∥}.

It is enough to show that x_{⊥} ∈ F^{⊥}:

(e_{m}|x_{⊥}) = (e_{m}|x) − (e_{m}| ∑_{n}(e_{n}|x)e_{n}) = 0.

b) Let x = x_{∥} + x_{⊥} and x = x′_{∥} + x′_{⊥}, then

F ∋ x′_{∥} − x_{∥} = x_{⊥} − x′_{⊥} ∈ F^{⊥} ⇒ x′_{∥} − x_{∥} = 0.
□

Corollary 2.3.1 If F is a closed subspace, then F^{⊥⊥} = F.

\underline{Proof} It is enough to show that F^{⊥⊥} ⊂ F. Let x ∈ F^{⊥⊥}, then by the Theorem 2.3.1 we have

x = x_{∥} + x_{⊥} x_{∥} ∈ F, x_{⊥} ∈ F^{⊥},

therefore

0 = (x|x_{⊥}) = (x_{∥}|x_{⊥}) + (x_{⊥}|x_{⊥}),

which gives

(x_{⊥}|x_{⊥}) = 0 ⇒ x_{⊥} = 0 ⇒ x ∈ F.
□

2.4

Proposition 2.4.1 a) G^{⊥⊥} is the smallest closed subspace containing the set G.

b) Let F be a subspace, then F is dense if and only if F^{⊥} = {0}.

□

3 Linear functionals

3.1

Definition 3.1.1 A linear (antilinear) functional is any linear (antilinear) map l : H → C.

Proposition 3.1.1 A linear functional l is continuous if and only if

∃c > 0 ∀x ∈ H |l(x)| ≤ c∥x∥.

□

3.2

Theorem 3.2.1 (Frechet, Riesz) For every continuous linear functional l there exists a vector y ∈ H such that

l(x) = (y|x) ∀x ∈ H.

\underline{Proof} Let {e_{n}} be an o.n. basis in H. First we shall show, that the series

∑_{n} \overline{l(e_{n})}e_{n}

is convergent:

∑_{n} |l(e_{n})|^{2} = ∑_{n} \overline{l(e_{n})}l(e_{n}) = l(∑_{n} \overline{l(e_{n})}e_{n}) ≤

≤ c∥ ∑_{n} \overline{l(e_{n})}e_{n}∥ ≤ c (∑_{n} |l(e_{n})|^{2})^{\frac{1}{2}} ,

hence

∑_{n} |l(e_{n})|^{2} ≤ c^{2}.

Now we can put

y =^{df} ∑_{n} \overline{l(e_{n})}e_{n}

and calculate

(y|e_{n}) = (∑_{m} \overline{l(e_{m})}e_{m}|e_{n}) = l(e_{n}).

□

Remark 3.2.1 It is very easy to formulate an analog of this theorem for continuous antilinear functionals.

4 Barrelled spaces

4.1

Definition 4.1.1 A subset Q ⊂ H is called absolutely convex if

∀x, y ∈ Q ∀α, β ∈ C (|α|^{2} + |β|^{2} ≤ 1) ⇒ (αx + βy ∈ Q) .

Definition 4.1.2 Q ⊂ H is called absorbing if ⋃_{n=1}^{∞} nQ = H.

Definition 4.1.3 A closed, absolutely convex, and absorbing set is called a barrel.

Definition 4.1.4 A topological vector space is called barrelled if every barrel is a neighbourhood of zero.

Proposition 4.1.1 A Hilbert space is barrelled.

□

4.2

Proposition 4.2.1 Let R be a subset in H such that

∀x ∈ H ∃M > 0 ∀y ∈ R |(x|y)| < M.

Then

∃N > 0 ∀y ∈ R ∥y∥ < N.

\underline{Proof} Let R^{o} =^{df} {z ∈ H : ∀y ∈ R |(z|y)| ≤ 1}. It is easy to see that R^{o} is closed and absolutely convex. We shall show that it is absorbing: If x ∈ H, then \frac{1}{M} x ∈ R^{o}. Thus x ∈ MR^{o} ⊂ nR^{o} for some n ∈ N. Hence R^{o} is a barrel, so it is a neighbourhood of zero i.e. there exists r > 0 such that the closed ball K(0, r) ⊂ R^{o}. Let y ∈ R. Since ∥\frac{ry}{∥y∥}∥ = r, \frac{ry}{∥y∥} ∈ R^{o}, which gives

|(\frac{ry}{∥y∥}|y)| = r∥y∥ ≤ 1,

therefore

∥y∥ ≤ \frac{1}{r}.

□

5 Bounded operators

5.1

Definition 5.1.1 A linear map A : H_{1} → H_{2} between two Hilbert spaces is called a bounded operator if

∃M ≥ 0 ∀x ∈ H_{1} ∥Ax∥_{2} ≤ M∥x∥_{1}.  (2)

The space of all bounded operators beetwen H_{1} and H_{2} will be denoted by B(H_{1}, H_{2}). Moreover, we define B(H) =^{df} B(H, H).

Proposition 5.1.1 a) If dim H_{1} < ∞, then every linear map is bounded.

b) Every bounded operator is continuous.

□

5.2

Of course, B(H_{1}, H_{2}) is a complex vector space and, in addition, it has a natural metric structure:

∥A∥ =^{df} sup_{∥x∥≤1} \frac{∥Ax∥_{2}}{∥x∥_{1}} ,

so ∥A∥ is the smalest M which satisfies the condition (2). It is not difficult to show that in the topology generated by this norm B(H_{1}, H_{2}) is complete (i.e. it is a Banach space), moreover, for B(H) we have the following inequality for the superposition of operators:

∥A · B∥ ≤ ∥A∥ · ∥B∥.

Definition 5.2.1 Let {A_{n}} be a sequence in B(H_{1}, H_{2}). Then we say:

a) {A_{n}} is convergent if {∥A_{n}∥} is convergent;

b) {A_{n}} is strongly convergent if for all x ∈ H {∥A_{n}x∥} is convergent;

c) {A_{n}} is weakly convergent if for all x, y ∈ H {(x|A_{n}y)} is convergent.

5.3

Proposition 5.3.1 For every A ∈ B(H_{1}, H_{2}) there exists exactly one A^{∗} ∈ B(H_{2}, H_{1}) such that

∀x ∈ H_{1}, ∀y ∈ H_{2} (y|Ax)_{2} = (A^{∗}y|x)_{1}.

\underline{Proof} For y ∈ H_{2} we have the following linear map:

H_{1} ∋ x ⟼ (y|Ax) ∈ C.  (3)

Because of

|(y|Ax)| ≤ ∥y∥∥Ax∥ ≤ ∥y∥∥A∥∥x∥,

the functional (3) is continuous, so by the Theorem 3.2.1 there exists z_{y} ∈ H_{1} such that (y|Ax) = (z_{y}|x) for all x ∈ B(H_{1}). Now we can define a linear map

H_{2} ∋ y ⟼ A^{∗}y =^{df} z_{y} ∈ H_{1}.

A^{∗} is linear:

(A^{∗}(αy_{1} + βy_{2})|x) = (αy_{1} + βy_{2}|Ax) =
= ᾱ(y_{1}|Ax) + β̄(y_{2}|Ax) =
= ᾱ(A^{∗}y_{1}|x) + β̄(A^{∗}y_{2}|x) =
= (αA^{∗}y_{1} + βA^{∗}y_{2}|x),

A^{∗} is continuous:

∥A^{∗}y∥^{2} = (A^{∗}y|A^{∗}y) = (y|AA^{∗}y) ≤
≤ ∥y∥∥A∥∥A^{∗}y∥,

therefore

∥A^{∗}y∥ ≤ ∥y∥∥A∥,

hence

∥A^{∗}∥ ≤ ∥A∥.
□

Definition 5.3.1 The operator A^{∗} is called an adjoint operator of A.

5.4

The following proposition show the elementary properties of the map

∗ : B(H_{1}, H_{2}) → B(H_{2}, H_{1}).

Proposition 5.4.1 a) A^{∗∗} = A, so ∗ is an involution;

b) ∥A^{∗}∥ = ∥A∥;

c) (λA + μB)^{∗} = λ̄A^{∗} + μ̄B^{∗}, so ∗ is antilinear;

d) (AB)^{∗} = B^{∗}A^{∗};

e) ∥A^{∗}A∥ = ∥A∥^{2}.

\underline{Proof}

a) (y|A^{∗∗}x) = (A^{∗}y|x) = (y|Ax),

b) from the proof of the Proposition 5.3.1 we know that ∥A^{∗}∥ ≤ ∥A∥, but ∥A∥ = ∥A^{∗∗}∥ ≤ ∥A^{∗}∥,

e) first, we have

∥A^{∗}A∥ ≤ ∥A^{∗}∥∥A∥ = ∥A∥^{2},

second, we have

∥A∥^{2} = sup_{∥x∥≤1} ∥Ax∥^{2} = sup_{∥x∥≤1}(Ax|Ax) = sup_{∥x∥≤1}(x|A^{∗}Ax) ≤

≤ sup_{∥x∥≤1} ∥x∥∥A^{∗}Ax∥ ≤ sup_{∥x∥≤1} ∥x∥∥A^{∗}A∥∥x∥ ≤ ∥A^{∗}A∥.

□

5.5

Definition 5.5.1 In the case H_{1} = H_{2} = H we define

a) if A = A^{∗}, then A will be called a Hermitian operator;

b) if for all x ∈ H (x|Ax) ≥ 0, then A will be called a positive operator;

c) if AA^{∗} = A^{∗}A, then A will be called a normal operator.

Proposition 5.5.1 Every positive operator is Hermitian.

□

5.6 Projection operators

Let as consider a closed subspace F ⊂ H. Then by the Theorem 2.3.1, for all x ∈ H we have x = x_{∥} + x_{⊥}, where x_{∥} ∈ F and x_{⊥} ∈ F^{⊥}. Because this decomposition is unique, we may define the following linear operator:

H ∋ x ↦ P_{F} x =^{df} x_{∥} ∈ H.

Proposition 5.6.1 a) P_{F} ∈ B(H), i.e. P_{F} is continuous;

b) P_{F}^{2} = P_{F}, i.e. P_{F} is idempotent;

c) P_{F}^{∗} = P_{F}, i.e. P_{F} is Hermitian.

\underline{Proof}

a) ∥P_{F} x∥ = ∥x_{∥}∥ ≤ ∥x_{∥} + x_{⊥}∥ = ∥x∥,

c) (y|P_{F} x) = (y_{∥} + y_{⊥}|P_{F} (x_{∥} + x_{⊥})) = (y_{∥} + y_{⊥}|x_{∥}) = (y_{∥}|x_{∥}) = = (y_{∥}|x_{∥} + x_{⊥}) = (P_{F} (y_{∥} + y_{⊥})|x_{∥} + x_{⊥}) = (P_{F} y|x). □

Proposition 5.6.2 If P ∈ B(H) and P^{2} = P = P^{∗}, then there exists a closed subspace F ⊂ H such that P = P_{F}.

\underline{Proof} We define a closed subspace in H:

F =^{df} {x ∈ H, P x = x}.

For every x ∈ H we have x = P x + (I − P)x, where I is the identity operator. We show that (I − P)x ∈ F^{⊥}: Let f ∈ F then

(f|(I − P)x) = ((I − P)^{∗}f|x) = ((I − P)f|x) = (f − P f|x) = (0|x) = 0.
□

Proposition 5.6.3 If F, G, F + G are projection operators, then F G = GF = 0.

\underline{Proof}

F + G = (F + G)^{2} = F^{2} + F G + GF + G^{2} = F + G + F G + F G,

which gives that F G + GF = 0. Multiplying this by F we obtain:

0 = F (F G + GF)F = F GF + F GF = 2F GF = 2F G^{2}F = 2(GF)^{∗}GF.

Because ∥A^{∗}A∥ = ∥A∥^{2}, we conclude that F G = 0. □

Proposition 5.6.4 Let E_{1}, ..., E_{n} ∈ B(H) be a family of projection operators such that E_{i} ≠ 0, E_{i}E_{j} = 0 for i ≠ j and ∑_{i=1}^{n} E_{i} = I. Then for λ_{1}, …, λ_{n} ∈ C

∥ ∑_{i=1}^{n} λ_{i}E_{i}∥ = max{|λ_{1}|, …, |λ_{n}|}.

\underline{Proof} For all x ∈ H, x = ∑_{i=1}^{n} E_{i}x and every component of this sum is orthogonal to each other. Moreover, if ∥x∥ ≤ 1 then ∑_{i=1}^{n} ∥E_{i}x∥ ≤ 1. Now using the notation a_{i} =^{df} ∥E_{i}x∥^{2} we may calculate:

∥ ∑_{i=1}^{n} λ_{i}E_{i}∥^{2} = sup_{∥x∥≤1} ∥ ∑_{i=1}^{n} λ_{i}E_{i}x∥^{2} = sup_{∥x∥≤1} ∑_{i=1}^{n} |λ_{i}|^{2}∥E_{i}x∥^{2} =

= sup ; a_{i}≥0,∑ a_{i}≤1 ∑_{i=1}^{n} |λ_{i}|^{2}a_{i} = max{|λ_{1}|^{2}, …, |λ_{n}|^{2}}.

□

5.7 Isometric and unitary operators

Definition 5.7.1 A linear operator U : H_{1} → H_{2} is called isometric if

∀x, y ∈ H_{1} (x|y)_{1} = (Ux|Uy)_{2},

if additionally U is ”onto,” then we call it a unitary operator.

Proposition 5.7.1 For every unitary operator U : H_{1} → H_{2} we have:

a) U ∈ B(H_{1}, H_{2}) and ∥U∥ = 1;

b) U^{∗}U = I_{1} – identity operator in H_{1};

c) UU^{∗} = I_{2} – identity operator in H_{2}.

□

Proposition 5.7.2 If U ∈ B(H_{1}, H_{2}), U^{∗}U = I_{1} and UU^{∗} = I_{2} then U is unitary.

\underline{Proof} Obviously U is an isometric operator. To show that for all z ∈ H_{2} there exists x ∈ H_{1} such that Ux = z we put x = U^{∗}z. □

Corollary 5.7.1 Operator U is unitary if and only if U^{∗} = U^{−1}.

□

6 Unbounded operators

6.1

The operator Af =^{df} \frac{df}{dt} acting in L^{2}([0, 1], dλ) is not an operator in the sense of the Definition 5.1.1, therefore we need more general definition:

Definition 6.1.1 A linear map T from H_{1} to H_{2} defined on a linear subspace D_{T} ⊂ H_{1} (T : H_{1} ⊃ D_{T} → H_{2}) is called an operator. D_{T} is called a domain of the operator T.

Definition 6.1.2 If T, S are operators from H_{1} to H_{2} such that D_{T} ⊂ D_{S} and for all x ∈ D_{T} T x = Sx, then we say that operator S is the extension of T and we write T ⊂ S.

6.2

Definition 6.2.1 An operator T is closed if for all sequences {x_{n}} ⊂ D_{T} we have

(x_{n} → x, T x_{n} → y) => (x ∈ D_{T} , T x = y) .

Definition 6.2.2 An operator T is closable if for all sequences {x_{n}} ⊂ D_{T} we have

(x_{n} → 0 T x_{n} → y) => (y = 0).

From every closable operator we can obtain a closed operator:

Definition 6.2.3 If T is a closable operator from H_{1} to H_{2}, then we define:

D\overline{T} =^{df} \{x ∈ H_{1} : ∃\{x_{n}\} ⊂ D_{T} , \{x_{n}\} & \{T x_{n}\} are convergent\} ,

\overline{T} x =^{df} lim_{n→∞} T x_{n}.

\overline{T} is called a closure of T. Of course T ⊂ \overline{T}.

6.3

Definition 6.3.1 For every operator T from H_{1} to H_{2} we define a subspace

G_{T} =^{df} {(x, T x) : x ∈ D_{T} } ⊂ H_{1} ⊕ H_{2},

which is called a graph of the operator T.

Proposition 6.3.1 A subset G ⊂ H_{1} ⊕ H_{2} is a graph of some operator T if and only if G is a vector subspace and G ∩ (H_{1} ⊕ {0}) = {0}.

Sketch of the proof

D_{T} =^{df} {x ∈ H_{1} : ∃y ∈ H_{2} (x, y) ∈ G},

T x =^{df} y.

□

Proposition 6.3.2 T is closed if and only if G_{T} is closed.

□

Proposition 6.3.3 T is closable if and only if \overline{G_{T}} (the closure of G_{T}) is a graph of some operator.

(\underline{Proof}) / (=>)

T ⊂ \overline{T} <=> G_{T} ⊂ G\overline{T} => \overline{G_{T}} ⊂ G\overline{T} => \overline{G_{T}} ∩ (H_{1} ⊕ \{0\}) = \{0\}.

<= Let as consider a sequence \{x_{n}\} ⊂ D_{T} such that x_{n} → 0 and T x_{n} → y, then (0, y) ∈ \overline{G_{T}}, hence y = 0 which means that T is closable. □

6.4

Analogically to bounded operators we want to introduce the notion of an adjoint operator. For the operator T : H_{1} ⊃ D_{T} → H_{2} we want to define an operator T^{∗} which satisfies conditions:

T^{∗} : H_{2} ⊃ D_{T^{∗}} → H_{1},
∀x ∈ D_{T} ∀y ∈ D_{T^{∗}} (x|T^{∗}y)_{1} = (T x|y)_{2}.  (4)

Since these two conditions are satisfied by T^{∗} = 0, D_{T^{∗}} = {0}, we need some kind of expansion.

Definition 6.4.1 Let T : H_{1} ⊃ D_{T} → H_{2} and D_{T} be dense in H_{1} i.e. \overline{D_{T}} = H_{1}. Then the operator T^{∗} adjoint to T may be defined by:

D_{T^{∗}} =^{df} {y ∈ H_{2} : ∃x ∈ H_{1} ∀x′ ∈ D_{T} (x′|x)_{1} = (T x′|y)_{2} } ,

T^{∗}y =^{df} x.

Remark 6.4.1 If D_{T} is not dense, then the equations (x′|x)_{1} = (T x′|y)_{2} do not determine x unique, so T^{∗} cannot exists.

From now on we shall consider only operators with dense domains.

Definition 6.4.2 Let V : H_{1} ⊕ H_{2} → H_{2} ⊕ H_{1} be given by:

V (x, y) =^{df} (−y, x).

Remark 6.4.2 V is a unitary operator.

Proposition 6.4.1 The condition (4) is equivalent to the inclusion:

G_{T^{∗}} ⊂ V G_{T}^{⊥}.

\underline{Proof} It is enough to show that V^{−1}G_{T^{∗}} is orthogonal to G_{T}: For x ∈ D_{T} and y ∈ D_{T^{∗}} we have V^{−1}(x, T^{∗}y) = (T^{∗}y, −x). Thus, the condition of orthogonality has the form (x|T^{∗}y)_{1} + (T x| − y)_{2} = 0, which is exactly (4). □

Proposition 6.4.2 Let T be an operator. D_{T} is dense in H_{1} if and only if V G_{T}^{⊥} is a graph.

\underline{Proof} (0, x) ∈ V G_{T}^{⊥} <=> (x, 0) ∈ G_{T}^{⊥} <=> x⊥D_{T} => x = 0 because D_{T} is dense. □

Corollary 6.4.1 We have an equivalent definition of the adjoint operator T^{∗}: T^{∗} is represented by the graph V G_{T}^{⊥}.

□

Corollary 6.4.2 For every operator T such that \overline{D_{T}} = H_{1} the operator T^{∗} is closed.

□

Proposition 6.4.3 a) If the operator T is closable, then D_{T^{∗}} is dense;

b) if D_{T^{∗}} is dense, then T is closable and T^{∗∗} = \overline{T}.

\underline{Proof}

a) Let y⊥D_{T^{∗}}, so (y, 0)⊥G_{T^{∗}}. Because V is unitary,

(y, 0) ∈ G_{T^{∗}}^{⊥} = V G_{T}^{⊥⊥} = V \overline{G_{T}} ,  (5)

therefore (0, y) ∈ \overline{G_{T}}. Because T is closable, then \overline{G_{T}} is a graph, hence y = 0.

b) from the formula (5) we obtain

(0, y) ∈ \overline{G_{T}} <=> y⊥D_{T^{∗}}.

Since D_{T^{∗}} is dense, y = 0. Thus, \overline{G_{T}} is a graph.

G_{T^{∗∗}} = V^{−1}G_{T^{∗}}^{⊥} = V^{−1}(V G_{T}^{⊥})^{⊥} = V^{−1}V G_{T}^{⊥⊥} = \overline{G_{T}} = G\overline{T} .
□

Corollary 6.4.3 If the operator T is closed, then T = T^{∗∗}.

□

Definition 6.4.3 Let H_{1} = H_{2}. Then:

a) if T ⊂ T^{∗}, T is called symmetric;

b) if T = T^{∗}, T is called self-adjoint;

c) if \overline{T} = T^{∗}, T is called essentially self-adjoint.

Corollary 6.4.4 T is symmetric if and only if for all x, y ∈ D_{T}

(T x|y) = (x|T y).

□

6.5

The following theorem shows why the domain of an unbounded operator cannot be an entire Hilbert space:

Theorem 6.5.1 If T is a closed operator from H_{1} to H_{2} and D_{T} = H_{1} then T is bounded.

\underline{Proof} Let P_{i}, i = 1, 2, denote the orthogonal projection in H_{1} ⊕ H_{2} onto the i-th component. Of course G_{T} = {(x, T x) : x ∈ H_{1}} is a Hilbert space and P_{1} |_{G_{T}}: G_{T} → H_{1} is an isomorphism of Hilbert spaces. Thus

x ⟼^{(P_{1}|_{G_{T}} )^{−1}} (x, T x) ⟼^{P_{2}} T x

being the superposition of continuous maps, is continuous. □

6.6

We denote by C(H_{1}, H_{2}) a set of all closed operators with the dense domain. If H_{1} = H_{2} = H, we write C(H). Note that C(H_{1}, H_{2}) is not a vector space, because, for example, T +(−T) ≠ 0, but T +(−T) = 0 |_{D_{T}}.

Theorem 6.6.1 If T ∈ C(H), then T^{∗}T is self-adjoint.

\underline{Proof} Since T is closed, G_{T} is a Hilbert subspace of H ⊕ H. So we have the orthogonal decomposition:

H ⊕ H = G_{T} ⊕ G_{T}^{⊥} = G_{T} ⊕ V^{−1}G_{T^{∗}}.  (6)

Thus, for every (x, 0) ∈ H ⊕ H there exists u ∈ D_{T} and v ∈ D_{T^{∗}} such that

(x, 0) = (u, T u) + V^{−1}(v, T^{∗}v) = (u − T^{∗}v, T u + v)

and therefore

x = u − T^{∗}v and v = −T u,

hence T u ∈ D_{T^{∗}} and u ∈ D_{T^{∗}T}, which allows us to write −T^{∗}v = T^{∗}T u. So we proved that

∀x ∈ H ∃u ∈ D_{T^{∗}T} x = u + T^{∗}T u.  (7)

Now we shall show that D_{T^{∗}T} is dense in H: Let x⊥D_{T^{∗}T}. Then, by (7), we have

0 = (x|u) = (u|u) + (T^{∗}T u|u) = (u|u) + (T u|T u) ≥ 0,

which implies u = 0, therefore x = 0.

It is easy to show that T^{∗}T is symmetric:

(x|T^{∗}T y) = (T x|T y) = (T^{∗}T x|y).

Finally, we shall show that D_{(T^{∗}T)^{∗}} ⊂ D_{T^{∗}T}:

Let x ∈ D_{(T^{∗}T)^{∗}} and y =^{df} (T^{∗}T)^{∗}x. Then for each h ∈ D_{T^{∗}T}

(y|h) = ((T^{∗}T)^{∗}x|h) = (x|T^{∗}T h).  (8)

From (7) we know that there exists u ∈ D_{T^{∗}T} such that x + y = u + T^{∗}T u.

(x − u|h + T^{∗}T h) = (x|h) − (u|h) + (x|T^{∗}T h) − (u|T^{∗}T h) =
= (x|h) − (u|h) + (y|h) − (T^{∗}T u|h) = (x + y|h) − (u + T^{∗}T u|h) = 0,

but h + T^{∗}T h runs over whole H (formula (7)) so x − u = 0. This means that x ∈ D_{T^{∗}T}. □

7 Spectral measures and integrals

7.1 σ–algebras

Let X be any set and B a family of subsets of X.

Definition 7.1.1 B ≠ ∅ is called a σ–algebra of sets if:

a) if A ∈ B, then X \ A ∈ B;

b) if A_{i} ∈ B, i ∈ N, then ∪_{i}A_{i} ∈ B.

Definition 7.1.2 The σ–algebra of subsets generated by B_{0} is the smallest σ–algebra of sets which contains B_{0}.

Definition 7.1.3 Let X be a topological space. Then

a) the σ–algebra of Borel sets is the σ–algebra generated by the family of all open sets;

b) the σ–algebra of Baire sets is the σ–algebra generated by the family of all compact sets of the G_{δ} type (countable intersection of open sets).

Remark 7.1.1 a) If X is a metric space, then every compact set is of the G_{δ} type;

b) (σ–algebra of Borel sets) ⊂ (σ–algebra of Baire sets);

c) if X = ∪_{n=1}^{∞}K_{n}, where K_{n} is compact, then (σ–algebra of Borel sets) = (σ–algebra of Baire sets).

□

7.2 Measures

Definition 7.2.1 Let (X, B) be a set with a σ–algebra of sets. A measure is any map

μ : B ⟶ \overline{R_{+}} =^{df} [0, ∞],

which satisfies the condition:

(A_{i} ∈ B, i ∈ N A_{i} ∩ A_{j} = ∅, i ≠ j) => (μ(∪_{i=1}^{∞}A_{i}) = ∑_{i=1}^{∞} μ(A_{i})) .

If additionally μ(X) = 1, then μ is called a probabilistic measure.

Definition 7.2.2 Let as consider (X, B). The function f : X → R is called B–measurable if

∀c ∈ R f^{−1}(] − ∞, c]) ∈ B.

Proposition 7.2.1 The measurable functions have the following properties:

a) they form an algebra (with ordinary pointwise addition and multiplication);

b) if {f_{n}} is a sequence of measurable functions and for all x ∈ X, f_{n}(x) → f(x), then f is measurable.

□

Definition 7.2.3 Baire(X) is, by definition, the set of all functions which are measurable with respect to the σ–algebra of Baire sets.

Proposition 7.2.2 C_{0}(X) ⊂ Baire(X), or, more precisely Baire(X) is the smallest family of functions which contains C_{0}(X) and is closed with respect to pointwise limits.

□

7.3 Integrals

Let as consider a set X with a σ–algebra of sets B and a measure μ.

Definition 7.3.1 Let χ_{A} be the characteristic function of the set A ∈ B. If μ(A) < ∞, then we define:

∫_{X} χ_{A}(x)μ(dx) =^{df} μ(A).

Definition 7.3.2 A function f is called a B–measurable stair function if μ({x : f(x) ≠ 0}) < ∞ and the value set of f is finite.

We see that f is a linear combination of characteristic functions:

f = ∑_{i=1}^{n} α_{i}χ_{A_{i}},

thus, we can define the integral

∫_{X} f(x)μ(dx) =^{df} ∑_{i=1}^{n} α_{i}μ(A_{i}).

The set of all B–measurable stair functions forms a vector structure and the integral is a positive functional on this space, so we can apply the Stone procedure to expand the integral on L^{p}(μ) and to construct L^{p}(μ) etc. Let as recall the following theorem from the general theory of the integral:

Theorem 7.3.1 Let X be a locally compact topological space and let ϕ : C_{0}(X) → R be a linear map which preserves positivity (i.e. if for all x ∈ X f(x) ≥ 0 then ϕ(f) ≥ 0). Then there exists a measure μ on the σ–algebra of Baire sets such that

ϕ(f) = ∫_{X} f(x)μ(dx).

□

Below we shall consider measures not necessarily positive i.e. the measures of the form μ_{1} − μ_{2} or μ_{1} + iμ_{2} (complex), where μ_{1}, μ_{2} are positive.

7.4 Spectral measures

Definition 7.4.1 Let (X, B) be a set with a σ–algebra of sets and let B(H) be the space of all bounded operators in a Hilbert space H. A spectral measure is any map E : B → B(H) which satisfies the following conditions:

a) for all A ∈ B, E(A) is a projection operator;

b) if A_{i} ∈ B, A_{i} ∩ A_{j} = ∅, i, j ∈ N, i ≠ j, x ∈ H, then

E (∪_{i=1}^{∞}A_{i}) x = ∑_{i=1}^{∞} E(A_{i})x;

c) E(X) = I is the identity operator.

Remark 7.4.1 Note that in the condition b) we have a strong convergence, not an operator convergence.

Proposition 7.4.1 Let E be a spectral measure and let A, B ∈ B. Then

E(A)E(B) = E(B)E(A) = E(A ∩ B).

\underline{Proof} Let A′ = A − B, B′ = B − A and C′ = A ∩ B. We have

E(A) = E(A′) + E(C′), E(B) = E(B′) + E(C′).

Now, by the Proposition 5.6.3 we can calculate:

E(A)E(B) = E^{2}(C′) = E(C′) = E(A ∩ B).
□

Remark 7.4.2 If E : B → B(H) is a spectral measure and x ∈ H, then the map B ∋ A ↦ (x|E(A)x) ∈ \overline{R_{+}} is a measure, and we denote it by (x|Ex).

7.5 Spectral integrals

Definition 7.5.1 Let (X, B, E) be a set with a σ–algebra of sets and a spectral measure. For the given stair function f(x) = ∑_{i=1}^{n} α_{i}χ_{A_{i}}(x), we define the spectral integral

∫_{X} f(x)E(dx) =^{df} ∑_{i=1}^{n} α_{i}E(A_{i}).

Proposition 7.5.1 The spectral integral has the following properties:

a) if f ≡ 1 then ∫_{X} f(x)E(dx) = I;

b) for all α, β ∈ C and f_{1}, f_{2} being stair functions

∫_{X}(αf_{1} + βf_{2})(x)E(dx) = α ∫_{X} f_{1}(x)E(dx) + β ∫_{X} f_{2}(x)E(dx);

c) for all f_{1}, f_{2} being stair functions

∫_{X} f_{1}(x)f_{2}(x)E(dx) = (∫_{X} f_{1}(x)E(dx)) (∫_{X} f_{2}(x)E(dx));

d) ∫_{X} \overline{f(x)}E(dx) = (∫_{X} f(x)E(dx))^{∗};

e) ∥ ∫_{X} f(x)E(dx)∥ ≤ sup_{x∈X} |f(x)|.

\underline{Proof}

b) Taking into account the definition and the fact that the integral does not depend on the representation of the stair function, i.e.

(∑_{i=1}^{n} α_{i}χ_{A_{i}} = ∑_{j=1}^{m} β_{j}χ_{B_{j}} ) => (∑_{i=1}^{n} α_{i}E(A_{i}) = ∑_{j=1}^{m} β_{j}E(B_{j})) ,

the proof is obvious. c) For f_{1} = ∑_{i=1}^{n} α_{i}χ_{A_{i}} and f_{2} = ∑_{j=1}^{m} β_{i}χ_{B_{j}} we have

f_{1}f_{2} = ∑_{i,j} α_{i}β_{j}χ_{A_{i}}χ_{B_{j}} = ∑_{i,j} α_{i}β_{j}χ_{A_{i}∩B_{j}},

which gives

∫_{X} f_{1}(x)f_{2}(x)E(dx) = ∑_{i,j} α_{i}β_{j}E(A_{i} ∩ B_{j}) = ∑_{i,j} α_{i}β_{j}E(A_{i})E(B_{j}).

e) The proof follows directly from the Proposition 5.6.4. □

Corollary 7.5.1 If {f_{n}} is a sequence of stair functions which is uniformly convergent to the function f then it follows from e) that the sequence ∫_{X} f_{n}(x)E(dx) is convergent and its limit depends only on f.

□

Definition 7.5.2 If {f_{n}} is a sequence of stair functions which is uniformly convergent to the function f, then we define

∫_{X} f(x)E(dx) =^{df} lim_{n→∞} ∫_{X} f_{n}(x)E(dx).

Proposition 7.5.2 Every bounded B–measurable function f may be conformly aproximated with stair functions.

\underline{Proof} Since f is bounded we may cover the set f(X) by ∪_{i=1}^{n}Δ_{i}, where |Δ_{i}| < ε. Now, defining:

A_{i} =^{df} {x ∈ X : f(x) ∈ Δ_{i}},

and

f_{ε} =^{df} ∑_{i=1}^{n} χ_{A_{i}},

we have

sup_{x∈X} |f(x) − f_{ε}(x)| ≤ ε.

□

Theorem 7.5.1 Let X be a compact space and let ϕ : C(X) → B(H) be a map which satisfies the following conditions:

a) ϕ is linear;

b) ϕ is multiplicative i.e. ϕ(f_{1}f_{2}) = ϕ(f_{1})ϕ(f_{2});

c) ϕ is symmetric i.e. ϕ(f̄) = (ϕ(f))^{∗};

d) ϕ(1) = I. Then there exists a spectral measure E on the σ–algebra of Baire sets such that for all f ∈ C(X)

ϕ(f) = ∫_{X} f(x)E(dx).

\underline{Proof} Step 1. Let M =^{df} sup_{x∈X} |f(x)|. Then M^{2} − f̄f ≥ 0, so there exists a nonnegative function g such that

M^{2} = f̄f + ḡg.

Acting on this equality by ϕ, we obtain

M^{2}I = ϕ(f)^{∗}ϕ(f) + ϕ(g)^{∗}ϕ(g)

and taking the average value over v ∈ H we have

M^{2}(v|v) = (v|ϕ(f)^{∗}ϕ(f)v) + (v|ϕ(g)^{∗}ϕ(g)v) =
= (ϕ(f)v|ϕ(f)v) + (ϕ(g)v|ϕ(g)v) = ∥ϕ(f)v∥^{2} + ∥ϕ(g)v∥^{2}.

Finally we obtain

∥ϕ(f)∥ ≤ sup_{x∈X} |f(x)|.

Step 2. If f ≥ 0 then ϕ(f) ≥ 0. Since because f = f̄^{\frac{1}{2}} f^{\frac{1}{2}} , ϕ(f) = ϕ(f^{\frac{1}{2}} )^{∗}ϕ(f^{\frac{1}{2}} ). Step 3. For each u ∈ H we define the map

C(X) ∋ f ↦ (u|ϕ(f)u) ∈ C.

This map assigns positive numbers to positive functions, so by the Theorem 7.3.1 there exists a measure μ_{u} such that

(u|ϕ(f)u) = ∫_{X} f(x)μ_{u}(dx).

Using the formula

(u|Av) = \frac{1}{4} ∑_{k=0}^{3} i^{k}(i^{k}u + v|A(i^{k}u + v)),

we obtain a complex measure μ_{u,v}, which satisfies

(u|ϕ(f)v) = ∫_{X} f(x)μ_{u,v}(dx).

Step 4. Now we may extend ϕ on a whole family of bounded Baire functions, denoted by Baire_{b}(X). Because X is compact, 1 ≡ f ∈ L^{1}(μ_{u,v}), so for all f ∈ Baire_{b}(X) we define

(u|ϕ̃(f)v) =^{df} ∫_{X} f(x)μ_{u,v}(dx).

As an exercise we propose to prove that this definition is good i.e.

i) the right-hand side of this definition is antilinear in u and continous in v, ii) ϕ̃ fulfil the conditions a)–d). For example, b) follows from the Lebesgue theorem on majority convergence. Step 5. We define the map

B ∋ A ↦ E(A) =^{df} ϕ̃(χ_{A}) ∈ B(H).

We prove that this is a spectral measure:

i) E(A) is a projection operator:

E(A)^{2} = ϕ̃(χ_{A})ϕ̃(χ_{A)} = ϕ̃(χ_{A}^{2}) = ϕ̃(χ_{A}) = E(A),
E(A)^{∗} = ϕ̃(χ_{A})^{∗} = ϕ̃(χ̄_{A}) = ϕ̃(χ_{A}) = E(A),

ii) E(X) = ϕ̃(1) = ϕ(1) = I, iii) countable additivity of E in strong topology: Let A = ∪_{i=1}^{∞}A_{i} where A_{i} ∩ A_{j} = ∅ for i ≠ j. For each u ∈ H we define a sequence {w_{N}} by

w_{N} =^{df} ∑_{i=1}^{N} E(A_{i})u.

To show that this sequence is convergent, it should be noted that the series

∑_{i=1}^{∞} ∥E(A_{i})u∥^{2}

is convergent. It follows from the fact that partial sums for this series are bounded by ∥u∥^{2}. Since the vectors E(A_{i})u are orthogonal, we have

∥w_{N} − w_{M}∥^{2} = ∥ ∑_{i=N+1}^{M} E(A_{i})u∥^{2} = ∑_{i=N+1}^{M} ∥E(A_{i})u∥^{2},

so {w_{N}} is convergent. Finally, from the fact that for all x ∈ X

χ_{A}(x) = lim_{N→∞} ∑_{i=1}^{N} χ_{A_{i}}(x),

we have

(u|E(A)v) = ∑_{i=1}^{∞}(u|E(A_{i})v),

which gives

lim_{N→∞} w_{N} = ∑_{i=1}^{∞} E(A_{i})u = E(∪_{i=1}^{∞}A_{i})u.

Step 6. Finally, we have that for every stair function f

ϕ̃(f) = ∫_{X} f(x)E(dx)  (9)

and, by the continuity, this is true for every f ∈ Baire_{b}(X). □

Definition 7.5.3 (Spectral integral for arbitrary Baire function.) Let f ∈ Baire(X) and {f_{n}} be a sequence in Baire_{b}(X) such that for all x ∈ X

|f_{n}(x)| ≤ |f(x)| f_{n}(x) → f(x).

We define an unbounded operator by

∫_{X} f(x)E(dx)u =^{df} lim_{n→∞} ∫_{X} f_{n}(x)E(dx)u,

with a domain consisting of those u ∈ H for which there exists the limit on the right side of this definition.

7.6

Directly from the definitions of spectral measure and integral we obtain:

Proposition 7.6.1 If ϕ : X_{1} → X_{2} is a measurable mapping and E is a spectral measure on X_{1}, then

F (Δ) =^{df} E(ϕ^{−1}(Δ))

is spectral measure on X_{2} and

∫_{X_{2}} f(x_{2})F (dx_{2}) = ∫_{X_{1}} f(ϕ(x_{1}))E(dx_{1}).

□

8 Commutative C^{∗}–algebras

8.1

Definition 8.1.1 A is a C^{∗}–algebra if:

a) A is a vector space over C;

b) there is in A a bilinear and associative multiplication

A × A ∋ (a, b) ↦ ab ∈ A,

(if additionally there exists 1 ∈ A such that for all a ∈ A 1a = a1 = a, we call A the algebra with identity);

c) there exists an antylinear map A ∋ a ↦ a^{∗} ∈ A such that for all a, b ∈ A

a^{∗∗} = a, (ab)^{∗} = b^{∗}a^{∗},

(it follows from this that 1^{∗} = 1);

d) A is a normed space such that for all a, b ∈ A

∥ab∥ ≤ ∥a∥∥b∥, ∥a^{∗}∥ = ∥a∥;

e) A is complete in this norm;

f) for all a ∈ A

∥a^{∗}a∥ = ∥a∥^{2};

g) if additionally for all a ∈ A, ab = ba, then A is called a commutative C^{∗}–algebra.

Definition 8.1.2 Element a ∈ A is called invertible if there exists a^{−1} ∈ A, such that aa^{−1} = a^{−1}a = 1. In opposite case, a is called noninvertible.

8.2 Examples

a) the set of bounded operators in a Hilbert space is a C^{∗}–algebra;

b) if A ∈ H and A^{∗} = A, then the closure of the space of all polinomials of A is a commutative C^{∗}–algebra;

c) if Λ is a compact topological space, then C(Λ) with

f^{∗} =^{df} f̄, ∥f∥ =^{df} sup_{λ∈Λ} |f(λ)|,

is a commutative C^{∗}–algebra. It turns aut that every commutative C^{∗}–algebra is of this form, more precisly, we have the following theorem which will be proved in this section:

Theorem 8.2.1 (Gelfand) Let A be a commutative C^{∗}–algebra. There exists a compact space Λ and a map A ∋ a ↦ â ∈ C(Λ) such that for all a, b ∈ A, r, s ∈ C and λ ∈ Λ:

a) ˆ (ra + sb)(λ) = râ(λ) + sb̂(λ);

b) âb(λ) = â(λ)b̂(λ);

c) â^{∗}(λ) = \overline{â(λ)};

d) ∥a∥ = sup_{λ∈Λ} |â(λ)|;

e) {â : a ∈ A} = C(Λ). Shortly, we have an isomorphism of C^{∗}–algebras.

Definition 8.2.1 A character of the C^{∗}–algebra A is any map A ∋ a ↦ χ(a) ∈ C such that for all a, b ∈ A and r, s ∈ C:

a) χ(ra + sb) = rχ(a) + sχ(b);

b) χ(ab) = χ(a)χ(b);

c) χ(a^{∗}) = \overline{χ(a)};

d) χ(1) = 1.

Remark 8.2.1 The notion of character is important because we shall define Λ as the space of all characters of A equipped in some topology and we shall put:

â(χ) =^{df} χ(a).

From now on A will denote a commutative C^{∗}–algebra with identity.

8.3 Ideals

Definition 8.3.1 A subset I ⊂ A is called an ideal if:

a) I is a vector subspace in A;

b) for all a ∈ I and b ∈ A we have ab ∈ I;

c) 1 ∉ I (i.e. I ≠ A).

Corollary 8.3.1 If I is an ideal, then A/I is an algebra, where

(a + I) + (b + I) =^{df} a + b + I,

(a + I)(b + I) =^{df} ab + I.

□

Proposition 8.3.1 Every element of each ideal is noninvertible and every noninvertible element in A is in some ideal.

\underline{Proof} Let a, a^{−1} ∈ I, then 1 ∈ I, which is a contradiction. For noninvertible a ∈ A we define the ideal

I =^{df} {ab : b ∈ A}.

□

8.4 Maximal ideals

Definition 8.4.1 M is called a maximal ideal if:

a) M is an ideal;

b) if there exists an ideal I, such that M ⊂ I, then M = I.

Proposition 8.4.1 Each ideal is included in some maximal ideal.

\underline{Proof} Follows from the Kuratowski – Zorn lema. □

Corollary 8.4.1 If a ∈ A is noninvertible, then there exists a maximal ideal M such that a ∈ M.

□

Proposition 8.4.2 If M is a maximal ideal, then A/M ≃ C.

\underline{Proof} Follows directly from the Gelfand–Mazur theorem, which will be proved later. □

Corollary 8.4.2 The map A ∋ a ↦ a + M ∈ A/M ≃ C is a character.

□

8.5

Proposition 8.5.1 If ∥1 − a∥ < 1, then a is invertible.

\underline{Proof} Let as consider the series ∑_{n=0}^{∞}(1 − a)^{n}. It is convergent, so by putting a = 1 − (1 − a) we see that this series is the inversion of a. □

Proposition 8.5.2 The closure of an ideal is an ideal too.

\underline{Proof} Let as consider an ideal I, then its closure Ī is a vector subspace in A. Let a = lim a_{n} ∈ Ī, a_{n} ∈ I, then for all b ∈ A we have ba = lim ba_{n} ∈ Ī. 1 ∉ Ī follows from the Proposition 8.5.1. □

Corollary 8.5.1 Every maximal ideal is closed.

□

8.6

Definition 8.6.1 a) The spectrum of an element a ∈ A is the set

Sp a =^{df} {λ ∈ C : a − λ1 is noninvertible };

b) the resolvent set of a:

ϱ(a) =^{df} C \ Sp a.

Proposition 8.6.1 a) If λ ∈ C is such that |λ| > ∥a∥, then λ ∈ ϱ(a);

b) ϱ(a) is open in C;

c) the map

ϱ(a) ∋ λ ↦ R(a, λ) =^{df} (a − λ1)^{−1} ∈ A

is holomorphic. R(a, λ) is called the resolvent of a;

d) Sp a ≠ ∅.

\underline{Proof}

a) Since |λ| > ∥a∥, the series

−\frac{1}{λ} ∑_{n=0}^{∞} \frac{a^{n}}{λ^{n}}  (10)

is convergent. Moreover

(a − λ1) (−\frac{1}{λ} ∑_{n=0}^{∞} \frac{a^{n}}{λ^{n}} ) = − ∑_{n−0}^{∞} \frac{a^{n+1}}{λ^{n+1}} + ∑_{n=0}^{∞} \frac{a^{n}}{λ^{n}} = 1.

Thus we have

R(a, λ) = (a − λ1)^{−1} = −\frac{1}{λ} ∑_{n=0}^{∞} \frac{a^{n}}{λ^{n}} .

b) Let λ_{0} ∈ ϱ(a), e.i. (a − λ_{0}1)^{−1} exists. Let as consider λ from some neighbourhood of λ_{0}. Then

(a − λ1)^{−1} = ((a − λ_{0}1) + (λ_{0} − λ)1)^{−1} =
= {(a − λ_{0}1) [1 − (λ − λ_{0})(a − λ_{0}1)^{−1}]}^{−1} =

= (a − λ_{0}1)^{−1} ∑_{n=0}^{∞} [(a − λ_{0}1)^{−1}(λ − λ_{0})]^{n} .

This series is convergent if |λ − λ_{0}| < ∥a − λ_{0}1∥^{−1}. Finally we see that each point in ϱ(a) has a neighbourhood in this set.

c) The proof is obvious because R(a, λ) is given by the series (10).

d) Assuming that Sp a = ∅, we have R(a, λ) defined on the whole C as a holomorphic function, which by (10) satisfies

R(a, λ) → 0 if |λ| → ∞.

Thus R(a, λ) ≡ 0 which is contradiction. □

Corollary 8.6.1 a) Sp a is compact;

b) we have the following equality

sup_{λ∈Sp a} |λ| = ∥a∥.

\underline{Proof}

b) Because the series (10) is convergent, then

lim_{n→∞} ^{n} \sqrt{\frac{∥a^{n}∥}{|λ|^{n}}} < 1,

which implies that

sup_{λ∈Sp a} |λ| = lim_{n→∞} ^{n} \sqrt{∥a^{n}∥}

is the radius of the minimal circle containing Sp a. But ∥a^{∗}a∥ = ∥a∥^{2}, so

∥a^{2}∥^{2} = ∥(a^{∗})^{2}a^{2}∥ = ∥a^{∗}aa^{∗}a∥ = ∥a^{∗}a∥^{2} = ∥a∥^{4},

therefore ∥a^{2}∥ = ∥a∥^{2} and

∥a^{2^{k}}∥ = ∥a∥^{2^{k}}.
□

Another corollary from the Proposition 8.6.1 is the following theorem

Theorem 8.6.1 (Gelfand, Mazur) If every a ∈ A \ {0} is invertible, then

A = {λ1 : λ ∈ C}.

\underline{Proof} For a ∈ A \ {0} there exists λ ∈ Sp a such that a − λ1 is noninvertible, so a − λ1 = 0. □

8.7

Let f be an entire function on C i.e. f(z) = ∑_{n=0}^{∞} c_{n}z^{n} and this series is convergent for all z ∈ C. For all a ∈ A we define

f(a) =^{df} ∑_{i=0}^{∞} c_{n}a^{n}.

It is easy to show that

f(a) = −\frac{1}{2πi} ∮ f(ζ)(a − ζ1)^{−1}dζ,

where the integral is taken over the contour including Sp a, moreover

Sp f(a) = f(Sp a).

Proposition 8.7.1 If a = a^{∗}, then Sp a ⊂ R.

\underline{Proof} Let as consider

b =^{df} e^{ia} = ∑_{n=0}^{∞} \frac{(ia)^{n}}{n!} .

We have b^{∗}b = e^{−ia}e^{ia} = e^{0} = 1, so ∥b∥ = 1, which gives

Sp b ⊂ {λ ∈ C : |λ| ≤ 1}.

But Sp b = e^{iSp a}, so

(z ∈ Sp a) => (Imz ≥ 0),

moreover Sp b^{∗} = e^{−iSp a}, so

(z ∈ Sp a) => (Imz ≤ 0).
□

8.8

Proposition 8.8.1 If a ∈ A and M is a maximal ideal in A, then there exists exactly one complex number λ_{M} such that

a − λ_{M}1 ∈ M.

\underline{Proof} Let as consider the C^{∗}–algebra A/M with the norm given by

∥a + M∥ =^{df} inf_{a′∈a+M} ∥a′∥.

Because M is maximal, then A/M satisfies the conditions of the Gelfand – Mazur theorem, thus

A/M ≃ C.

□

Definition 8.8.1 Let M denote the set of all maximal ideals in A. We define the map called the Gelfand transformation

A ∋ a ↦ â ∈ {complex functions on M},

given by

â(M) =^{df} λ_{M}.

Theorem 8.8.1 For all a, b ∈ A:

a) ˆ (a + b) = â + b̂;

b) âb = âb̂;

c) ˆ1 = 1 – constant function on M;

d) â(M) = Sp a;

e) defining

∥â∥ =^{df} sup_{M∈M} |â(M|,

we have that ∥a∥ = ∥â∥;

f) â^{∗} = ¯ â;

g) â(M) = 0 if and only if a ∈ M.

\underline{Proof}

b) ab − â(M)b̂(M)1 = (a − â(M)1)b + â(M)(b − b̂(M)1) ∈ M.

d) if λ ∈ Sp a, then there exists a maximal ideal M such that a − λ1 ∈ M, so λ = â(M). Thus Sp a ⊂ â(M). Now let λ ∈ â(M) which means that there exists a maximal ideal M such that λ = â(M), so a − λ1 is noninvertible i.e. λ ∈ Sp a. Thus â(M) ⊂ Sp a. e) we have

∥â∥ = sup_{M∈M} |â(M)| = sup_{λ∈Sp a} |λ| = ∥a∥.

f) the proof follows from the observation that

b = b^{∗} => b̂(M) ∈ R

and that a = \frac{1}{2}(a + a^{∗}) + i\frac{1}{2i}(a − a^{∗}) and a^{∗} = \frac{1}{2}(a + a^{∗}) − i\frac{1}{2i}(a − a^{∗}). □

8.9 Topology on M

To prove the spectral theorem, it is enough to limit our considerations to the case when A is generated by a finite number of elements a_{1}, ..., a_{n} ∈ A i.e. A is the closure of the space of all polinomials of a_{i}, i = 1, …, n.

Proposition 8.9.1 If M_{1}, M_{2} ∈ M and â_{i}(M_{1}) = a_{i}(M_{2}), i = 1, ..., n, then M_{1} = M_{2}.

\underline{Proof} Follows from the fact that

M = {a ∈ A : â(M) = 0}.
□

Corollary 8.9.1 We have the injection

M ∋ M ↦ (a_{1}(M), …, a_{n}(M)) ∈ C^{n}.  (11)

□

Definition 8.9.1 The topology on M is defined as a pull-back, by the map (11), of the usual topology from C^{n}.

Theorem 8.9.1 a) M is compact;

b) for all a ∈ A the function â is continuous;

c) each continous function on M is of the form â, for some a ∈ A.

\underline{Proof}

a) M is bounded because |â_{i}(M)| ≤ ∥a_{i}∥. We shall show that M is closed: Let {M_{k}} be a Couchy sequence in M, so there exists (λ_{1}, …, λ_{n}) ∈ C^{n} such that for all i = 1, …, n,

lim_{k→∞} â_{i}(M_{k}) = λ_{i}.

We must show that there exists the maximal ideal M_{∞} such that â_{i}(M_{∞}) = λ_{i} for all i. It follows from the proposition 8.9.1 that for all a ∈ A there exists λ_{a} ∈ C such that

λ_{a} = lim_{k→∞} â(M_{k}).

Thus we have the multiplicative and linear map

A ∋ a ↦ λ_{a} ∈ C,

which leads to the definition:

M_{∞} =^{df} {a ∈ A : λ_{a} = 0}.

It is easy to see that M_{∞} is an ideal. Because codim M_{∞} = 1, we obtain that M_{∞} is a maximal ideal. Let z_{i} =^{df} â_{i}(M_{∞}), then a_{i} − z_{i}1 ∈ M_{∞} i.e.

0 = λ_{a_{i}−z_{i}1} = λ_{i} − z_{i}.

Finally we have

lim_{k→∞} M_{k} = M_{∞} ∈ M.

b) All â_{i} are continuous, so all its polinomials and limits of polinomials (because convergence in A implies uniform convergence of the Gelfand transformations).

c) The proof follows directly from the Stone–Weierstrass theorem. □

Remark 8.9.1 In this way we completed the proof of the Gelfand theorem 8.2.1.

9 Spectral theorems

9.1 Spectral theorem for Hermitian operators

Let as assume that A is a C^{∗}–subalgebra in B(H) generated by a Hermitian operator A.

Theorem 9.1.1 If A ∈ B(H) is a Hermitian operator, then there exists a spectral measure E on Sp A, such that

A = ∫_{Sp A} λE(dλ).

\underline{Proof} The inverse Gelfnad trasformation is an example of an isomorphism between C(M) and A which satisfies the conditions of the Theorem 7.5.1. This implies that there exists a spectral measure E′ on M such that for all B ∈ A we have

B = ∫_{M} B̂(λ)E′(dλ),

where B̂ is the Gelfand transformation of B. Moreover, due to the Theorem 8.8.1 d) and the Proposition 8.9.1, we may identify M with Sp A ⊂ R. Taking into account the Proposition 7.6.1, we see that there exists a spectral measure E on Sp A such that

B = ∫_{Sp A} B̂(Â_{−1}(λ))E(dλ),  (12)

where Â_{−1} is the inverse map to the Gelfand transformation of the generator of A, i.e. Â_{−1} : Sp A → M. Putting A in (12) we complete the proof. □

This proof show that we may also formulate the spectral theorem for unitary operators or, more generally for normal operators. If an operator A is not normal, then we cannot imbed A into some commutative algebra.

9.2 Cayley transformation

Now let T be an unbounded operator in a Hilbert space H and let D_{T} denote the domain of T.

Proposition 9.2.1

(T = T^{∗}) => ((T ± iI)D_{T} = H).

\underline{Proof} Step 1. We shall prove that (T ± iI)D_{T} is dense in H: Let as assume that there exist 0 ≠ u⊥(T ± iI)D_{T}, which means that for all v ∈ D_{T} we have (u|(T ± iI)v) = 0, so (u|T v) = (±iu|v), which implies that u ∈ D_{T^{∗}} and T^{∗}u = iu. But T^{∗} = T, so T u = iu, which iscontradiction. Step 2. We shall prove that (T ± iI)D_{T} is closed in H: Because T = T^{∗}, then for each v ∈ D_{T} we can directly calculate that

∥(T ± iI)v∥^{2} = ∥T v∥^{2} + ∥v∥^{2}.  (13)

Now let h ∈ H, then there exists a sequence {v_{n}} such that

(T + iI)v_{n} → h if n → ∞.

Thus, by (13), {v_{n}} and {T v_{n}} are Couchy sequences, so there exist v, w ∈ H such that

v_{n} → v and T v_{n} → w.

Moreover, T is closed (because T = T^{∗}), then T v = w. Finally (T + iI)v = h. □

Corollary 9.2.1 T ± iI is invertible.

\underline{Proof} It follows from (13) that ker(T ± iI) = {0}. □

Definition 9.2.1 The Cayley transformation of an operator T = T^{∗} is the operator

U_{T} =^{df} (T − iI)(T + iI)^{−1}.

Proposition 9.2.2 U_{T} has the following properties:

a) the domain of U_{T} is the whole H;

b) U_{T} (H) = H;

c) for all h ∈ H we have ∥U_{T} h∥ = ∥h∥.

□

Corollary 9.2.2 U_{T} is a unitary operator.

□

9.3 Idea of a proof of the spectral theorem for self-adjoint operators

In particular, we have the spectral representation of U_{T}:

U_{T} = ∫_{|λ|=1} λE_{U_{T}} (dλ).

The main idea is as follows: Using the spectral representation of U_{T} we want to obtain the representation of T due to the inverse Cayley transformation:

T = −i(U_{T} − I)^{−1}(U_{T} + I)

Thus we propose

T = ∫_{|λ|=1} −i\frac{λ + 1}{λ − 1}E_{U_{T}} (dλ),

where, of course

−i\frac{λ + 1}{λ − 1} ∈ R.

It is easy to check that the mapping

S^{1} − \{1\} ∋ λ ↦ C^{−1}(λ) =^{df} −i\frac{λ + 1}{λ − 1} ∈ R

is ”onto.” So we may define the spectral measure on R:

E_{T} (Δ) =^{df} E_{U_{T}} (C(Δ)).

Now, by the Proposition 7.6.1 we obtain:

T = ∫_{S^{1}} C^{−1}(λ)E_{U_{T}} (dλ) = ∫_{R} μE_{U_{T}} (C(dμ)) =

= ∫_{R} μE_{T} (dμ),

where μ = C^{−1}(λ).

9.4 The spectral theorem for self-adjoint operators

Theorem 9.4.1 If T is a self-adjoint operator in a Hilbert space H, then there exists a spectral measure E_{T} on the real line R such that:

T = ∫_{−∞}^{+∞} μE_{T} (dμ).

\underline{Proof} For u ∈ D_{T} we have (from the Cayley transformation) the following formula

U_{T} (T + iI)u = (T − iI)u  (14)

or, in another form

(U_{T} − I)T u = −i(U_{T} + I)u.  (15)

Let as consider the equation (U_{T} − I)v = 0. By the Proposition 9.2.1 let x ∈ D_{T} be such that v = (T + iI)x. Using (14) we obtain U_{T} v = (T − iI)x. But our equation gives U_{T} v = v. Therefore (T + iI)x = (T − iI)x, which means ix = −ix or x = 0, so v = 0. Concluding, the equation (15) determines T u in a unique manner.

It will be convenient to define

C_{ε} =^{df} {λ ∈ S^{1} : | arg λ| ≥ ε}

and

H_{ε} =^{df} E_{U_{T}} (C_{ε})H.

For u ∈ H_{ε} ∩ D_{T} we have a good defined bounded operator:

T u = ∫_{C_{ε}} −i\frac{λ + 1}{λ − 1}E_{U_{T}} (dλ)u,

which, for our u, we may write in the following form:

T u = ∫_{S^{1}} −i\frac{λ + 1}{λ − 1}E_{U_{T}} (dλ)u,

(of course, S^{1} = C_{0}).

We shall check that this formula satisfies the equation (15):

(U_{T} − I) ∫_{S^{1}} −i\frac{λ + 1}{λ − 1}E_{U_{T}} (dλ)u =

= ∫_{S^{1}}(λ − 1)E_{U_{T}} (dλ) ∫_{S^{1}} −i\frac{λ + 1}{λ − 1}E_{U_{T}} (dλ)u =

= −i ∫_{S^{1}}(λ + 1)E_{U_{T}} (dλ)u = −i(U_{T} + I)u.

This allows us to put for u ∈ D_{T}:

T u =^{df} lim_{ε→0} ∫_{C_{ε}} −i\frac{λ + 1}{λ − 1}E_{U_{T}} (dλ)u.  (16)

First of all we check that this T u satisfies (15) too: Since the operator U_{T} − I is continous, it commutes with the operation of taking limit

(U_{T} − I) lim_{ε→0} ∫_{C_{ε}} −i\frac{λ + 1}{λ − 1}E_{U_{T}} (dλ)u =

= lim_{ε→0} ∫_{C_{ε}} −i(λ + 1)E_{U_{T}} (dλ)u =^{(∗)}

= −i ∫_{C_{0}}(λ + 1)E_{U_{T}} (dλ)u = −i(U_{T} + I)u,

where the equality (*) follows from the fact that:

lim_{ε→0} ∥ ∫_{C_{0}\C_{ε}}(λ + 1)E_{U_{T}} (dλ)u∥^{2} = (u|E_{U_{T}} ({1})u) = 0,

because E_{U_{T}} ({1}) = 0 (i.e. there does not exist a vector 0 ≠ v ∈ H, such that U_{T} v = v).

To prove that the limit in (16) exists, it should be noted that we may rewrite (15) in the following form:

∫_{C_{0}}(λ − 1)E_{U_{T}} (dλ)T u = −i ∫_{C_{0}}(λ + 1)E_{U_{T}} (dλ)u.

Now, because ∫_{C_{ε}}(λ − 1)^{−1}E_{U_{T}} (dλ) a bounded operator,

∫_{C_{ε}} 1E_{U_{T}} (dλ)T u = ∫_{C_{ε}}(λ − 1)^{−1}E_{U_{T}} (dλ) ∫_{C_{0}}(λ − 1)E_{U_{T}} (dλ)T u =

= ∫_{C_{ε}} −i\frac{λ + 1}{λ − 1}E_{U_{T}} (dλ)u.

Thua the limit exists and by the Proposition 7.6.1 we have for all u ∈ D_{T}:

T u = ∫_{−∞}^{+∞} μE_{T} (dμ)u,

where μ = −i\frac{λ+1}{λ−1}. Thus we have

T ⊂ ∫_{−∞}^{+∞} μE_{T} (dμ).

It is easy to see that ∫_{−∞}^{+∞} μE_{T} (dμ) is symmetric, then, because T = T^{∗}, we have:

T = ∫_{−∞}^{+∞} μE_{T} (dμ).
□