INTRODUCTION TO SPECTRAL THEORY

WILLIAM CASPER

Abstract. This is an abstract.

1. Introduction

Throughout this paper, we let H denote a Hilbert space, X a measureable space and Ω a σ-algebra of subsets of X. By an operator T on H we will always mean a linear transformation such that the operator norm

∥T ∥ := sup{∥T ψ∥ : ψ ∈ H, ∥ψ∥ = 1}

is finite. An operator will be called invertible if it has an algebraic inverse T^{−1} which is also an operator on H.

1.1. Basic Definitions and Facts.

Definition 1.1. The spectrum of a linear operator T on H is

(1) spec(T) = {λ ∈ ℂ : T − λ is not invertible. }.

The approximate point spectrum of T is

aspec(T) = {λ ∈ ℂ : ϱ(T − λ) = 0}, where ϱ(T) = inf{∥T ψ∥ : ψ ∈ H, ∥ψ∥ = 1}. (2)

Theorem 1.1. aspec(T) ⊂ spec(T).

Theorem 1.2. If T is a normal operator, then aspec(T) = spec(T).

Theorem 1.3 (Transforms of Spectra).

(i) If p ∈ ℂ[x], then

spec(p(T)) = p(spec(T)) := {p(λ) : λ ∈ spec(T)}.

(ii) If T is invertible, then

spec(T^{−1}) = (spec(T))^{−1} := {λ^{−1} : λ ∈ spec(T)}.

(iii) The spectrum of the adjoint of T satisfies

spec(T^{∗}) = (spec(T))^{∗} := {λ^{∗} : λ ∈ spec(T)}.

Theorem 1.4. Define

N_{T} (f) = sup{|f(λ)| : λ ∈ spec(T)}. (3)

In general, spec(T) is a compact subset of the complex plane and N_{T} (x) ≤ ∥T ∥. If T is Hermitian, then spec(T) is a subset of ℝ and N_{T} (p(x)) = ∥p(T)∥ for any polynomial p ∈ ℝ[x].

Date: February 24, 2010.

2. Spectral Measures

Definition 2.1. A spectral measure E on a measureable space (X, Ω) is a projection-valued (idempotent, hermitian) set function

E : Ω → { projections on H}

satisfying

(i) E(X) = 1; (ii) for any collection of pairwise disjoint sets {A_{k}}_{k=1}^{∞},

E ( ⋃_{k=1}^{∞} A_{k}) = ∑_{k=1}^{∞} E(A_{k}).

Theorem 2.1 (Properties of Spectral measures). If E is a spectral measure, then for all A, B ∈ Ω

(i) if A ⊂ B, then E(A) ≤ E(B); (ii) if A ⊂ B, then E(B \ A) = E(B) − E(A); (iii) E(A ∪ B) + E(A ∩ B) = E(A) + E(B); (iv) E(A ∩ B) = E(A)E(B).

Theorem 2.2. A function

E : Ω → { projections on H}

is a spectral measure if and only if

(i) E(X) = 1; (ii) for any two fixed elements ψ, ϕ ∈ H,the function μ : Ω → ℂ defined by

μ(A) = ⟨E(A)ψ, ϕ⟩ for all A ∈ Ω

is a complex measure. We use the notation d⟨E(λ)ψ, ϕ⟩ = dμ(λ) so that in general

∫ 1_{A}d⟨E(λ)ψ, ϕ⟩ = ⟨E(A)ψ, ϕ⟩ for all A ∈ Ω.

Theorem 2.3. If E is a spectral measure and f is an E-measureable function, then there exists a unique operator denoted by either ∫ fdE or ∫ f(λ)dE(λ) and defined as

⟨∫ fdEψ, ϕ⟩ = ∫ f(λ)d⟨E(λ)ψ, ϕ⟩. (4)

Theorem 2.4 (Properties of ∫ fdE). Given any E-measureable functions f, g and α ∈ ℂ,

(i) ∫ (αf)dE = α ∫ fdE; (ii) ∫ (f + g)dE = ∫ fdE + ∫ gdE; (iii) (∫ fdE)^{∗} = ∫ f^{∗}dE; (iv) ∫ fgdE = (∫ fdE) (∫ fdE).

Theorem 2.5. If E is a spectral measure and E(A) commutes with T for every A ∈ Ω, then ∫ fdE commutes with T.

3. Complex Spectral Measures

For the remainder of the paper, we assume that X is a locally compact Hausdorff space and that Ω is the Borel σ-algebra on X.

Definition 3.1. A spectral measure is regular if for all A ∈ Ω,

E(A) = sup{E(C) : C ⊂ A, C is compact }.

Definition 3.2. The spectrum of a spectral measure is

spec(E) := X \{λ ∈ X : λ ∈ A, A is open, E(A) = 0}.

A spectral measure is compact if its spectrum is compact.

Theorem 3.1. If E is a regular spectral measure, then spec(E) is closed and E(X \ spec(E)) = 0 (and therefore E(spec(E)) = 1).

Theorem 3.2. For any complex-valued, E-measureable function bounded on spec(E), define

N_{E}(f) = sup{|f(λ)| : λ ∈ spec(E)}.

Then if E is a compact, regular spectral measure and f is a continuous function on X, then ∥∫ fdE∥ = N_{E}(f).

Definition 3.3. A spectral measure is called complex when X = ℂ.

Theorem 3.3. Every complex spectral measure is regular.

Theorem 3.4. If E is a compact, complex spectral measure and if T = ∫ λdE(λ), then spec(T) = spec(E).

Theorem 3.5. A complex spectral measure is defined completely by the operator ∫ λdE(λ). That is, given two complex spectral measures E_{1} and E_{2}, E_{1} = E_{2} if and only if ∫ λdE_{1}(λ) = ∫ λdE_{2}(λ).

Theorem 3.6. Let E be a complex spectral measure and T an operator. Then T commutes with E(A) for all A ∈ Ω if and only if T commutes with both ∫ λdE(λ) and ∫ λ^{∗}dE(λ).

4. The Spectral Theorem

Theorem 4.1 (Spectral Theorem for Hermitian Operators). Let T be a Hermitian operator. Then there exists a unique compact, complex spectral measure E such that T = ∫ λdE(λ).

Theorem 4.2 (Spectral Theorem for Normal Operators). Let T be a normal operator. Then there exists a unique compact, complex spectral measure E such that T = ∫ λdE(λ).

Definition 4.1. For any normal operator T, we call the spectral measure E satisfying T = ∫ λdE(λ) the spectral measure of T.

Appendix A. Applications of the Spectral Theorem

A.1. Weak Mixing. As a first example application of the spectral theorem, we will use it to show that a measure-preserving transformation T is weak mixing when the only eigenfunctions of the unitary operator U_{T} defined by U_{T} f(x) = f(T x) are the constants.

Definition A.1. Let (X, Ω, μ) be a probability space and T a measure preserving transformation (mpt) on X (μ(A) = μ(T^{−1}(A)) for all A ∈ Ω). Then T is called weakly mixing if

(5)  lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} |μ(T^{k}(A ∩ B) − μ(A)μ(B)| → 0

for all A, B ∈ Ω.

Theorem A.1. A mpt T is weakly mixing if the only measureable eigenfunctions of U_{T} are the constants.

Proof. We first note that T is weakly mixing if and only if

(6)  lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩ − ⟨f, 1⟩⟨1, g⟩| → 0

for all f, g ∈ L_{2}(μ).

Let V be the closed linear subspace of the eigenfunctions of T in L_{2}(μ) and let E be the spectral measure of T. Then for any λ_{0} ∈ spec(U_{T}), we have that

U_{T} E({λ_{0}}) = ∫ λdE(λ) ∫ 1_{λ_{0}}(λ)dE(λ) = ∫ λ1_{{λ_{0}}}(λ)dE(λ) = λ_{0}E({λ_{0}}),

so that in particular U_{T} E({λ_{0}})f = λ_{0}f for every f ∈ L_{2}(μ). Thus E({λ_{0}})f ∈ V for all f ∈ L_{2}(μ). If f ∈ V^{⊥}, this implies that

0 = ⟨E({λ_{0}})f, f⟩ = ⟨E({λ_{0}})^{2}f, f⟩ = ⟨E({λ_{0}})f, E({λ_{0}})f⟩,

and therefore E({λ_{0}})f = 0.

Now fix an f ∈ V^{⊥} and g ∈ L_{2}(μ) and define μ to be the complex Borel measure on the spectrum of T satisfying dμ = d⟨E(λ)f, g⟩. Then for all λ_{0} ∈ spec(T), we have that

μ({λ_{0}}) = ⟨E({λ})f, g⟩ = 0.

Setting Δ = {(λ, ω) ∈ spec(U_{T}) × spec(U_{T}) : λ = ω}, we find

\frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩|^{2} = \frac{1}{n} ∑_{k=0}^{n−1} |⟨∫ λ^{k}dE(λ)f, g⟩|^{2} = \frac{1}{n} ∑_{k=0}^{n−1} |λ^{k} ∫ ⟨dE(λ)f, g⟩|^{2}

= \frac{1}{n} ∑_{k=0}^{n−1} | ∫ λ^{k}dμ(λ)|^{2} = \frac{1}{n} ∑_{k=0}^{n−1} ∫ λ^{k}dμ(λ) ∫ (\overline{ω})^{k}dμ^{∗}(ω)

= \frac{1}{n} ∑_{k=0}^{n−1} ∫ ∫ (λ\overline{ω})^{k}dμ(λ)dμ^{∗}(ω) = ∫ \frac{1}{n} ∑_{k=0}^{n−1}(λ\overline{ω})^{k}d(μ × μ^{∗})(λ, ω)

= ∫_{Δ^{c}} \frac{1}{n} \frac{1 − (λ\overline{ω})^{k}}{1 − λ\overline{ω}} d(μ × μ^{∗})(λ, ω) + ∫_{Δ} 1d(μ × μ^{∗})(λ, ω).

Additionally,

∫_{Δ} 1d(μ × μ^{∗})(λ, ω) = ∫ ∫ 1_{Δ}(λ, ω)dμ(λ)dμ^{∗}(ω) = ∫ μ({ω})dμ^{∗}(ω) = 0.

For (λ, ω) ∉ Δ, (1 − (λ\overline{ω})^{k})/(1 − λ\overline{ω}) is a cyclotomic polynomial, and is therefore bounded on the compact set spec(U_{T}). Thus by the bounded convergence theorem

lim_{n→∞} ∫_{Δ^{c}} \frac{1}{n} \frac{1 − (λ\overline{ω})^{k}}{1 − λ\overline{ω}} d(μ × μ^{∗})(λ, ω) = ∫_{Δ^{c}} 0d(μ × μ^{∗})(λ, ω) = 0.

We conclude that

lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩|^{2} = 0.

By the Cauchy-Schwartz inequality,

(∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩|)^{2} ≤ n ∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩|^{2}.

Dividing both sides by n^{2}, we find

(\frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩|)^{2} ≤ \frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩|^{2},

and therefore

lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩| = 0.

Now for any f, g ∈ L_{2}(G), if the eigenfunctions of U_{T} are the constants, f − ⟨f, 1⟩ ∈ V^{⊥}. Therefore

lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}(f − ⟨f, 1⟩), g⟩| = lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}f − ⟨f, 1⟩, g⟩|

= lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} |⟨U_{T}^{k}f, g⟩ − ⟨f, 1⟩⟨1, g⟩| = 0.

Therefore T is weakly mixing. □

A.2. Almost Periodic Functions.

Appendix B. Proofs of Theorems

B.1. Proof for Section 1.

Proof of Theorem (1.1). If λ ∉ spec(T), then T − λ is invertible. Thus for any ψ ∈ H with ∥ψ∥ = 1, we have that

1 = ∥ψ∥ = ∥(T − λ)^{−1}(T − λ)ψ∥ ≤ ∥(T − λ)^{−1}∥ · ∥(T − λ)ψ∥.

Thus ϱ (as defined by Eq. (1)) satisfies ϱ(T − λ) ≥ 1/∥(T − λ)^{−1}∥ and so λ ∉ aspec(T). This proves our theorem. □

Lemma B.1. An operator T is invertible if and only if its range is dense in H and there exists a positive real number c > 0 such that ∥T ψ∥ ≥ c∥ψ∥ for all ψ ∈ H.

Proof. If T is invertible, then it is a bijection and therefore the range must be H. Moreover,

∥ψ∥ = ∥T^{−1}T ψ∥ ≤ ∥T^{−1}∥ · ∥T ψ∥,

so ∥T ψ∥ ≥ c∥ψ∥ with c = 1/∥T^{−1}∥.

Conversely, suppose the range of T is dense in H and there exists a positive real number c > 0 such that ∥T ψ∥ ≥ c∥ψ∥ for all ψ ∈ H. We first show that the range of T is H. Let {ϕ_{i}}_{i=1}^{∞} be a convergent sequence in the range of H converging to ϕ. For every i > 0, there exists a ψ_{i} ∈ H such that T ψ_{i} = ϕ_{i}. Moreover, ∥ψ_{i} − ψ_{j}∥ ≤ ∥ϕ_{i} − ϕ_{j}∥/c. It follows that the sequence {ψ_{i}}_{i=1}^{∞} is Cauchy and therefore converges to a function ψ ∈ H. Since T is continuous, ϕ = T ψ, and therefore ϕ is in the range of T. We conclude that the range of T is closed. Since the range of T is dense in H, the range of T must be H.

The kernel of T is trivial, since if ψ ∈ ker(T), then ∥ψ∥ ≤ ∥T ψ∥/c = 0, implying that ψ = 0. Thus T is a bijection, and all that is left to show is that the algebraic inverse, which we call T^{−1}, is bounded. We have that ∥T^{−1}ψ∥ ≤ ∥T T^{−1}ψ∥/c ≤ ∥ψ∥/c. It follows that ∥T^{−1}∥ ≤ 1/c. This proves our theorem. Incidentally, this also shows us that c = 1/∥T^{−1}∥ is the ”sharpest” value for c. □

Proof of Theorem (1.2). By Theorem (1.1), we need only prove spec(T) ⊂ aspec(T). Suppose that λ ∉ aspec(T). Then there exists a constant c > 0 such that ∥(T − λ)ψ∥ ≥ c∥ψ∥ for all ψ ∈ H. By Lemma (B.1), we need only show that the range of T − λ is dense in H. Since T commutes with T^{∗}, T − λ commutes with (T − λ)^{∗} = T^{∗} − λ^{∗}, and it follows that ∥(T − λ)ψ∥ = ∥(T^{∗} − λ^{∗})ψ∥ for all ψ ∈ H. If ϕ ∈ rangle(T − λ)^{⊥}, then 0 = ⟨(T − λ)ψ, ϕ⟩ = ⟨ψ, (T^{∗} − λ^{∗})ϕ⟩ for all ψ ∈ H, and therefore (T^{∗} − λ^{∗})ϕ = 0. It follows that ϕ = 0, since ∥ϕ∥ ≤ ∥(T − λ)ϕ∥/c = ∥(T^{∗} − λ^{∗})ϕ∥/c = 0. Thus rangle(T − λ)^{⊥} = {0} and it follows that rangle(T − λ) is dense in H. This proves our theorem. □

Proof of Theorem (1.3). (i) Let p ∈ ℂ[x] and λ ∈ spec(T). Then λ is a root of r(x) = p(x) − p(λ) and therefore there exists a polynomial q ∈ ℂ[x] such that q(x)(x − λ) = p(x) − p(λ). If r(T) is invertible, then q(T) commutes with r^{−1}(T) and

(T − λ)q(T)r(T)^{−1} = r(T)r(T)^{−1} = 1 = r(T)^{−1}r(T) = r(T)^{−1}(T − λ)q(T)
= r(T)^{−1}q(T)(T − λ) = q(T)r(T)^{−1}(T − λ).

It follows that (T − λ) is invertible with (T − λ)^{−1} = q(T)r(T)^{−1}, which is a contradiction. Thus r(T) is not invertible and p(λ) ∈ spec(p(T)).

Conversely, suppose λ ∈ spec(p(T)) and let {r_{i}}_{i=1}^{n} be the roots of the polynomial p(x) − λ. We have that p(x) − λ = (x − r_{1}) … (x − r_{n}) and therefore p(T) − λ = (T − r_{1}) … (T − r_{n}). Since p(T) − λ is not invertible, (T − r_{j}) is not invertible for some j. Whence r_{j} ∈ spec(T) and p(r_{j}) − λ = 0. We conclude that λ ∈ p(spec(T)). This proves (i).

(ii) Note that for any λ ∈ ℂ, we have that T^{−1} − λ^{−1} = −T^{−1}λ^{−1}(T − λ), and it follows that T^{−1} − λ^{−1} is invertible if and only if T − λ is invertible. This proves (ii).

(iii) If λ ∉ spec(T), then T − λ is invertible. It follows that (T − λ)^{∗} = T^{∗} − λ^{∗} is invertible, and therefore spec(T^{∗}) ⊂ spec(T)^{∗}. By the same argument with T replaced by T^{∗}, spec(T) = spec((T^{∗})^{∗}) ⊂ spec(T^{∗})^{∗}, and therefore spec(T)^{∗} ⊂ (spec(T^{∗})^{∗})^{∗} = spec(T^{∗}). This proves our theorem.

□

Lemma B.2. If T is an operator such that ∥1 − T ∥ < 1, then T is invertible.

Proof. Define c > 0 by c = 1 − ∥1 − T ∥. Then

∥T ψ∥ = ∥ψ − (ψ − T ψ)∥ ≥ ∥ψ∥ − ∥(1 − T)ψ∥ ≥ ∥ψ∥ − ∥(1 − T)∥ · ∥ψ∥ = c∥ψ∥.

Thus by Lemma (B.1), we need only show that the range of T is dense in H. Let ϕ ∈ H and let δ = inf{∥T ψ − ϕ∥ : ψ ∈ H}. Suppose that δ > 0. Then for all ϵ = δ \frac{c}{1−c}, there exists ψ ∈ H such that δ ≤ ∥T ψ − ϕ∥ < δ + ϵ. Moreover

δ ≤ ∥T (T ψ−ϕ)−(T ψ−ϕ)∥ = ∥(1−T)(T ψ−ϕ)∥ < (1−c)∥T ψ−ϕ∥ = (1−c)(δ+ϵ) ≤ δ.

That is, δ < δ, which is a contradiction. We conclude that δ = 0. Since ϕ ∈ H was taken arbitrarily, this means that the range of T is dense in H. This proves our lemma. □

Proof of Theorem (1.4). If λ_{0} ∉ spec(T), then T − λ_{0} is invertible. If λ ∈ ℂ with |λ − λ_{0}| < r := 1/∥(T − λ_{0})^{−1}∥, then

∥1 − (T − λ_{0})^{−1}(T − λ)∥ = ∥(T − λ_{0})^{−1}[(T − λ_{0}) − (T − λ)]∥
≤ ∥(T − λ_{0})^{−1}∥ · |λ − λ_{0}| < 1.

Therefore by Lemma (B.2) (T − λ_{0})^{−1}(T − λ) is invertible and it follows that (T − λ) must be invertible. We conclude that the ball B(λ_{0}; r) about λ_{0} of radius r is contained in ℂ \ spec(T). It follows that ℂ \ spec(T) is open and therefore spec(T) is closed. Moreover, if λ ∈ ℂ satisfies ∥T ∥ < |λ|, then ∥1 − (1 − T/λ)∥ = ∥T/λ∥ < 1 and therefore 1−T/λ is invertible by Lemma(B.2). It follows that T −λ is invertible, and therefore λ ∉ spec(T). Thus if λ ∈ spec(T), then |λ| ≤ ∥T ∥ necessarily. In particular, this shows that N_{T} (x) ≤ ∥T ∥ and that spec(T) is a closed and bounded subset of ℂ (and therefore compact).

Suppose T is Hermitian and λ ∈ spec(T). Then T is normal and spec(T) = aspec(T) by Theorem (1.2). Thus there exists a sequence {ψ_{i}}_{i=1}^{∞} ⊂ H such that ∥ψ_{i}∥ = 1 for all i and ∥(T − λ)ψ_{i}∥ → 0. Thus

|λ − λ^{∗}| = |λ − λ^{∗}| · ∥ψ_{i}∥^{2} = |⟨(T − λ)ψ_{i}, ψ_{i}⟩ − ⟨(T − λ^{∗})ψ_{i}, ψ_{i}⟩|
= |⟨(T − λ)ψ_{i}, ψ_{i}⟩ − ⟨ψ_{i}, (T − λ)ψ_{i}⟩|
≤ 2∥(T − λ)ψ_{i}∥ · ∥ψ_{i}∥ = 2∥(T − λ)ψ_{i}∥ → 0.

It follows that λ is real. Moreover for any λ ∈ ℝ, since T is Hermitian, we have the relation

∥T^{2}ψ − λ^{2}ψ∥^{2} = ⟨T^{2}ψ − λ^{2}ψ, T^{2}ψ − λ^{2}ψ⟩
= ∥T^{2}ψ∥^{2} + |λ|^{4}∥ψ∥^{2} − (λ^{2})^{∗}⟨T^{2}ψ, ψ⟩ − λ^{2}⟨ψ, T^{2}ψ⟩
= ∥T^{2}ψ∥^{2} + λ^{4}∥ψ∥^{2} − 2λ^{2}∥T ψ∥^{2}.

Now let {ψ_{i}}_{i=1}^{∞} ⊂ H be a sequence such that ∥ψ_{i}∥ = 1 for all i and ∥T ψ_{i}∥ → ∥T ∥. Then taking λ = ∥T ∥ in the above relation, we find that

∥(T^{2} − ∥T ∥^{2})ψ_{i}∥^{2} = ∥T^{2}ψ_{i} − λ^{2}ψ_{i}∥^{2} = ∥T^{2}ψ_{i}∥^{2} + λ^{4}∥ψ_{i}∥^{2} − 2λ^{2}∥T ψ_{i}∥^{2}
= ∥T^{2}ψ_{i}∥^{2} + ∥T ∥^{4} − 2∥T ∥^{2}∥T ψ_{i}∥^{2} → 0.

Thus ∥T ∥^{2} ∈ spec(T^{2}), and it follows from Theorem (1.3) that either ∥T ∥ ∈ spec(T) or −∥T ∥ ∈ spec(T). In particular, this proves N_{T} (x) = ∥T ∥. If p(x) ∈ ℝ[x], then p(T) is Hermitian and therefore N_{T} (p(x)) = N_{p(T)}(x) = ∥p(T)∥. This proves our theorem. □

B.2. Proofs for Section 2.

B.3. Proofs for Section 3.

B.4. Proofs for Section 4.

Lemma B.3 (Weierstrass Approximation Theorem). Let X be a compact subset of ℝ and let f be a continuous function on X. Then there exists a sequence of real polynomials {p_{i}}_{i=1}^{∞} such that p_{i} → f uniformly on X.

Lemma B.4. Let L be a bounded linear functional on R[x] and let X be a compact subset of R. Then there exists a unique Borel measure μ on X satisfying

L(p) = ∫ p(λ)dμ(λ) for all p ∈ ℝ[x].

Sketch of proof. Let Ω be the collection of all Borel subsets of X and let A ∈ Ω. Let {p_{i}}_{i=1}^{∞} ⊂ ℝ[x] be a sequence of polynomials with p_{i} → 1_{A} uniformly on X. Define μ(A) by

μ(A) = lim_{i→∞} L(p_{i}).

Then μ is a well-defined complex Borel measure on X. □

Proof of Theorem (4.1). Let ψ, ϕ ∈ H and define a function

L : ℝ[x] → ℂ

by L(p) = ⟨p(T)ψ, ϕ⟩. Then L is linear and

|L(p)| ≤ ∥p(T)ψ∥ · ∥ϕ∥ ≤ ∥p(T)∥ · ∥ψ∥ · ∥ϕ∥ ≤ N_{T} (p(x)) · ∥ψ∥ · ∥ϕ∥

and therefore L is a linear functional on ℝ[x]. The set ℝ[x] is a dense subset of the collection of all continuous, real-valued functions on spec(T), and it follows that there exists a unique complex measure μ on X = spec(T) with σ-algebra Ω consisting of all Borel subsets of spec(T) such that L(p) = ∫ p(λ)dμ(λ) for all p ∈ ℂ[x]. For given ψ, ϕ ∈ H, we denote this measure by μ_{(ψ,ϕ)}. Let ψ_{1}, ψ_{2}, ϕ_{1}, ϕ_{2} ∈ H and let α ∈ ℂ.

∫ p(λ)dμ_{(ψ_{1}+ψ_{2},ϕ)}(λ) = ⟨p(T)(ψ_{1} + ψ_{2}), ϕ⟩ = ⟨p(T)ψ_{1}, ϕ⟩ + ⟨p(T)ψ_{2}, ϕ⟩
= ∫ p(λ)dμ_{(ψ_{1},ϕ)}(λ) + ∫ p(λ)dμ_{(ψ_{2},ϕ)}(λ),

from which it follows that

μ_{(ψ_{1}+ψ_{2},ϕ)} = μ_{(ψ_{1},ϕ)} + μ_{(ψ_{2},ϕ)}.

Similarly,

μ_{(ψ,ϕ_{1}+ϕ_{2})} = μ_{(ψ,ϕ_{1})} + μ_{(ψ,ϕ_{2})};
μ_{(αψ,ϕ)} = αμ_{(ψ,ϕ)} and μ_{(ψ,αϕ)} = α^{∗}μ_{(ψ,ϕ)}.

Lastly, for A ∈ Ω we have that

|μ_{(ψ,ϕ)}(A)| ≤ |μ_{(ψ,ϕ)}|(X) = sup \{\frac{1}{N_{T} (p)} |∫ pdμ_{(ψ,ϕ)}| : p ∈ ℂ[x]\}
= sup {|⟨p(T)ψ, ϕ⟩|/N_{T} (p) : p ∈ ℂ[x]} = ∥ψ∥ · ∥ϕ∥.

For any A ∈ Ω, we define μ_{A}(ψ, ϕ) = μ_{(ψ,ϕ)}(A). The above properties show us that μ_{A} is a symmetric, bilinear functional, and therefore for every A ∈ Ω, there exists a unique Hermitian operator E(A) such that μ_{A}(ψ, ϕ) = ⟨E(A)ψ, ϕ⟩ for all ψ, ϕ ∈ H.

We first show that E(A) is idempotent for all A ∈ Ω by proving the more general result E(A ∩ B) = E(A)E(B) for all A, B ∈ Ω. Fix B ∈ Ω and let {q_{i}}_{i=1}^{∞} ⊂ ℝ[x] be a fixed sequence of polynomials with q_{i}(λ) → 1_{A}(λ) uniformly on X. Also fix ψ, ϕ ∈ H. For each i, define a measure ν_{i} by dν_{i}(λ) = q_{i}(λ)dμ_{(ψ,ϕ)}(λ). Then for any p ∈ ℂ[x], q(T) is Hermitian commutes with p(T) and we have that

∫ p(λ)dν_{i}(λ) = ∫ p(λ)q_{i}(λ)dμ_{(ψ,ϕ)}(λ) = ⟨p(T)q_{i}(T)ψ, ϕ⟩
= ⟨p(T)ψ, q_{i}(T)ϕ⟩ = ∫ p(λ)dμ_{(ψ,q_{i}(T)ϕ)}.

Let A ∈ Ω and {p_{i}}_{i=1}^{∞} ⊂ ℝ[x] be a sequence of polynomials with p_{i}(λ) → 1_{A}(λ) uniformly on X. Then the dominated convergence theorem tells us that

ν_{i}(A) = lim_{i→∞} ∫ p_{i}(λ)q_{i}(λ)dμ_{(ψ,ϕ)}(λ) = lim_{i→∞} ∫ p_{i}(λ)dμ_{(ψ,q_{i}(T)ϕ)}(λ)
= ∫ 1_{A}(λ)dμ_{(ψ,q_{i}(T)ϕ)}(λ) = μ_{A}(ψ, q_{i}(T)ϕ) = ⟨E(A)ψ, q_{i}(T)ϕ⟩
= ⟨q_{i}(T)E(A)ψ, ϕ⟩ = ∫ q_{i}(λ)dμ_{(E(A)ψ,ϕ)}(λ)

The dominated convergence theorem also tells us that

⟨E(A ∩ B)ψ, ϕ⟩ = ∫ 1_{A∩B}(λ)dμ_{(ψ,ϕ)}(λ) = ∫ 1_{A}(λ)1_{B}(λ)dμ_{(ψ,ϕ)}(λ)

= lim_{i→∞} ν_{i}(A) = lim_{i→∞} ∫ q_{i}(λ)dμ_{(E(A)ψ,ϕ)}(λ)

= ∫ 1_{B}(λ)dμ_{(E(A)ψ,ϕ)}(λ) = ⟨E(B)E(A)ψ, ϕ⟩.

Since A, B ∈ Ω and ψ, ϕ ∈ H were arbitrary, this proves that E(A∩B) = E(A)E(B) for all A, B ∈ Ω. Thus E is idempotent.

Lastly, we have that

⟨E(X)ψ, ϕ⟩ = μ_{X}(ψ, ϕ) = μ_{(ψ,ϕ)}(X) = ∫ 1dμ_{(ψ,ϕ)}(λ) = ⟨ψ, ϕ⟩,

and by Theorem (2.2) this means that E(X) is a compact, complex spectral measure. A quick calculation shows that

∫ λdμ_{(ψ,ϕ)}(λ) = ⟨T ψ, ϕ⟩.

The uniqueness of the measure follows from Theorem (3.5). This proves our theorem. □

Sketch of proof of Theorem (4.2). Let T be normal and define T_{1} and T_{2} by T_{1} = \frac{1}{2}(T + T^{∗}) and T_{2} = \frac{1}{2i}(T − T^{∗}). Then T_{1} and T_{2} are Hermitian with T = T_{1} + iT_{2} and there exist unique spectral measures E_{1}, E_{2} such that T_{i} = ∫ λdE_{i}(λ) for i = 1, 2.

Define A = {A + iB : A, B real Borel sets } and define a projection-valued set function E by E(A + iB) = E_{1}(A)E_{2}(B). Then A is an algebra of sets and the σ-algebra generated by A is Ω, the collection of all Borel subsets of ℂ. Moreover, E(ℂ) = E_{1}(ℝ)E_{2}(ℝ) = 1 and E extends uniquely to a projection-valued measure E on ℂ.

□