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Spectral theorem for bounded normal operators pvm form

Statement

Assume AC. Let $T$ be a bounded normal operator on a nonzero complex Hilbert space $H$ , with operator spectrum $σ (T)$ (Spectrum and resolvent of a bounded operator). Then:

there is a unique regular projection valued measure $E$ on the Borel $σ$ -algebra of the nonempty compact set $σ (T) \subseteq C$ such that $\int z d E (z) = T$ and, equivalently, $Φ_{E} (f) = f (T) for every continuous f : σ (T) \to C,$ where $f (T)$ is the continuous functional calculus of Continuous functional calculus for bounded normal operators and $Φ_{E}$ is the bounded Borel integral of the projection valued measure $E$ ; this $E$ is the spectral projection valued measure of $T$ ;
conversely, if $Λ \subseteq C$ is nonempty and compact and $E$ is a regular projection valued measure on the Borel $σ$ -algebra of $Λ$ , then $T_{E} : = \int z d E (z) = Φ_{E} (z)$ is a bounded normal operator with $σ (T_{E}) \subseteq Λ$ and $∥ T_{E} ∥ \leq \max_{z \in Λ} ∣ z ∣$ .

Facts & Assumptions

[A1]

For normal $T$ the map $π (f) : = f (T)$ is the unique isometric unital star-isomorphism $C (σ (T)) \to C^{*} (I, T)$ with $π (z) = T$ ; it is complex-linear, multiplicative, unital and star-preserving, and its range consists of the continuous-calculus operators (Continuous functional calculus for bounded normal operators, Continuous functional calculus properties, C star algebra generated by a normal operator).

[A2]

$σ (T)$ is a nonempty compact subset of $C$ : the operator spectrum coincides with the spectrum of $T$ in the unital C*-algebra $C^{*} (I, T)$ by spectral permanence and the bounded inverse theorem, and the spectrum of an element of a nonzero unital complex Banach algebra is nonempty and compact (Spectral permanence for unital c star subalgebras, Bounded inverse theorem, Spectrum is nonempty compact and norm bounded, Spectrum and resolvent of a bounded operator).

[A3]

For a nonempty compact Hausdorff $K$ and a unital star-homomorphism $π : C (K; C) \to B (H)$ there is a unique regular PVM $E$ on the Borel $σ$ -algebra of $K$ with $Φ_{E} (f) = π (f)$ for every continuous $f$ , and for bounded Borel $h$ the operator $Φ_{E} (h)$ satisfies $∥ Φ_{E} (h) ∥ \leq ∥ h ∥_{\infty}$ (Continuous functional calculus produces a regular PVM, Bounded borel pvm integral).

[A4]

For every PVM the map $Φ_{E}$ is linear, unital, multiplicative and star-preserving on bounded Borel functions; consequently $T_{E} : = Φ_{E} (z)$ is normal because $T_{E} T_{E}^{*} = Φ_{E} (∣ z ∣^{2}) = T_{E}^{*} T_{E}$ (Pvm integral is a star homomorphism).

[A5]

For an orthogonal projection value $E (Λ) = I$ and a continuous bounded $g$ on $Λ$ the operators $Φ_{E} (g)$ are bounded by $∥ g ∥_{\infty}$ , and for $∣ λ - z ∣ > 0$ on $Λ$ the function $z \mapsto (λ - z)^{- 1}$ is continuous (Bounded borel pvm integral, Projection valued measure).

[A6]

The $*$ -polynomials in $z, \overline{z}$ are uniformly dense in $C (Λ; C)$ for compact $Λ \subseteq C$ , and the image of a compact set under a continuous map is compact (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense, The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset).

[A7]

A bounded operator with a two-sided bounded inverse at $λ$ has $λ \notin σ (T)$ ; normality is $T^{*} T = T T^{*}$ (Spectrum and resolvent of a bounded operator, Self-adjoint, positive, unitary and normal operators, Hilbert space).

[A8]

AC is the declared choice hypothesis of this page from the construction item onward (The Axiom of Choice).

Proof

technique · direct

Given: A nonzero complex Hilbert space $H$ and a bounded normal operator $T \in B (H)$ ; for the converse a nonempty compact $Λ \subseteq C$ and a regular PVM $E$ on its Borel $σ$ -algebra.

1.1

Existence: $σ (T)$ is a nonempty compact subset of $C$ and $π (f) : = f (T)$ is a unital star-homomorphism $C (σ (T); C) \to B (H)$ ; applying the construction lemma to $K = σ (T)$ gives a regular PVM $E$ on the Borel $σ$ -algebra of $σ (T)$ with $Φ_{E} (f) = π (f) = f (T)$ for every continuous $f$ ; taking $f$ to be the coordinate function $z$ gives $\int z d E = π (z) = T$ .

A1A2A3

1.2

Converse, boundedness and normality: for a regular PVM $E$ on a nonempty compact $Λ$ the coordinate function is bounded by $\max_{z \in Λ} ∣ z ∣$ and measurable, so $T_{E} = Φ_{E} (z)$ is a bounded operator with $∥ T_{E} ∥ \leq \max_{z \in Λ} ∣ z ∣$ , its adjoint is $T_{E}^{*} = Φ_{E} (\overline{z})$ by conjugation preservation, and $T_{E} T_{E}^{*} = Φ_{E} (z \overline{z}) = Φ_{E} (\overline{z} z) = T_{E}^{*} T_{E}$ by multiplicativity, so $T_{E}$ is normal.

A4A5

2.1

Converse, spectrum: if $λ \in C ∖ Λ$ then $Λ$ is closed and $g (z) : = (λ - z)^{- 1}$ is continuous on $Λ$ with $(λ - z) g (z) = 1$ ; hence $(λ I - T_{E}) Φ_{E} (g) = Φ_{E} (λ 1 - z) Φ_{E} (g) = Φ_{E} ((λ - z) g) = Φ_{E} (1) = I$ and likewise $Φ_{E} (g) (λ I - T_{E}) = I$ , so $λ I - T_{E}$ has a two-sided bounded inverse and $λ \notin σ (T_{E})$ ; therefore $σ (T_{E}) \subseteq Λ$ .

step 1.2A4A5A7

2.2

Uniqueness for the direct statement: if $E^{'}$ is a regular PVM on the Borel $σ$ -algebra of $σ (T)$ with $\int z d E^{'} = T$ , then for every $*$ -polynomial $p (z, \overline{z})$ one has $Φ_{E^{'}} (p) = p (T_{E}^{'}, T_{E}^{' *}) = p (T, T^{*}) = p (T) = Φ_{E} (p)$ , using multiplicativity and conjugation preservation of $Φ_{E^{'}}$ together with $T_{E}^{'} = T$ and the identification of $T^{*}$ with the calculus value of $\overline{z}$ ; since $*$ -polynomials are uniformly dense in $C (σ (T))$ and both $Φ_{E^{'}}$ and $Φ_{E}$ are bounded linear maps agreeing there, they agree on every continuous function, so $E^{'} = E$ by the uniqueness clause of the construction lemma.

step 1.1A1A3A6

3.1

The spectral PVM $E$ of $T$ therefore exists, is unique among regular PVMs on $σ (T)$ whose coordinate integral is $T$ , and satisfies $Φ_{E} (f) = f (T)$ for all continuous $f$ ; conversely every coordinate integral of a regular PVM on a compact set is a bounded normal operator with spectrum inside that set.

step 1.1step 1.2step 2.1step 2.2A8∎

Depends on

Used by

A normal operator need not have any eigenvectors Counterexample
Borel functional calculus for a bounded normal operator Definition
Pvm of a diagonal normal operator Example
Pvm of a multiplication operator Example
Spectral projection of an isolated eigenvalue agrees with the riesz projection Example
Borel functional calculus for bounded normal operators Theorem
Spectral theorem for unbounded self-adjoint operators (PVM form) Theorem
Support and uniqueness of the spectral measure Theorem

Dependency tree · two levels

93 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.74, printed pp.288–291 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem, Corollary 5.7, pp.19–20 (standard reference, not scraped)