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Continuous functional calculus for bounded normal operators

Statement

Assume AC. For a bounded normal operator $T$ on a nonzero complex Hilbert space there is a unique isometric unital star-isomorphism $C (σ (T)) \to C^{*} (I, T)$ , $f \mapsto f (T)$ , sending the coordinate function $z$ to $T$ ; for self-adjoint $T$ it agrees with the self-adjoint calculus.

Facts & Assumptions

[A1]

For normal $T$ the nonzero unital commutative C*-algebra $C^{*} (I, T)$ has a nonempty compact Hausdorff character space, and the map $Φ : Δ (C^{*} (I, T)) \to σ (T)$ , $χ \mapsto χ (T)$ , is a homeomorphism onto the therefore nonempty compact spectrum; pullback $f \mapsto f \circ Φ$ is consequently an isometric bijection $C (σ (T)) \to C (Δ (C^{*} (I, T)))$ preserving pointwise sums, products and conjugation (Maximal ideal space is compact Hausdorff, Character space of generated normal algebra is operator spectrum, C star algebra).

[A2]

For a nonzero unital commutative complex C*-algebra $C$ the Gelfand transform $Γ : C \to C (Δ (C))$ is an isometric unital $*$ -isomorphism onto $C (Δ (C))$ , so its inverse has the same properties (Commutative Gelfand Naimark).

[A3]

For normal $T$ the generated algebra $C^{*} (I, T)$ is a nonzero unital commutative C*-algebra with the same identity as $B (H)$ (C star algebra generated by a normal operator).

[A4]

Every unital point-separating self-adjoint complex function algebra on a nonempty compact Hausdorff space is uniformly dense in the continuous functions; applied to $σ (T) \subseteq C$ this makes the $*$ -polynomials in $z$ and $\overline{z}$ uniformly dense in $C (σ (T))$ (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense).

[A5]

For self-adjoint $T$ there is a unique isometric unital star-homomorphism $C (σ (T)) \to B (H)$ with $z \mapsto T$ and range $C^{*} (I, T)$ (Continuous functional calculus for bounded self adjoint operators).

[A6]

Every self-adjoint operator is normal (Self-adjoint, positive, unitary and normal operators).

[A7]

AC is the hypothesis of the Gelfand and character-space suppliers (The Axiom of Choice).

Proof

technique · direct

Given: A nonzero complex Hilbert space $H$ and a bounded normal operator $T \in B (H)$ , with $Δ : = Δ (C^{*} (I, T))$ and $Φ (χ) = χ (T)$ .

1.1

The algebra $C^{*} (I, T)$ is a nonzero unital commutative C*-algebra with the same identity as $B (H)$ , its character space $Δ$ is nonempty compact Hausdorff, $Φ : Δ \to σ (T)$ is a homeomorphism, and the pullback map $U (f) : = f \circ Φ$ is an isometric bijection $C (σ (T)) \to C (Δ)$ that preserves pointwise sums, products and conjugation.

A1A3

1.2

The inverse Gelfand transform $Γ^{- 1} : C (Δ) \to C^{*} (I, T)$ is an isometric unital $*$ -isomorphism onto $C^{*} (I, T)$ .

2.1

The composite $Ψ : = Γ^{- 1} \circ U : C (σ (T)) \to C^{*} (I, T)$ is an isometric unital $*$ -isomorphism onto $C^{*} (I, T)$ , and $Ψ (z) = Γ^{- 1} (\hat{T}) = T$ because the Gelfand transform of $T$ is $\hat{T} (χ) = χ (T) = Φ (χ) = z (Φ (χ)) = (z \circ Φ) (χ)$ .

step 1.1step 1.2algebra

3.1

$Ψ$ is the only isometric unital $*$ -isomorphism $C (σ (T)) \to C^{*} (I, T)$ with $z \mapsto T$ : if $Ξ$ is another, then $Ξ^{- 1} \circ Ψ$ is a unital $*$ -isomorphism of $C (σ (T))$ fixing $z$ and $\overline{z}$ , hence fixing every $*$ -polynomial in $z$ ; these are uniformly dense by Stone–Weierstrass and the map is isometric, so it is the identity on $C (σ (T))$ and $Ξ = Ψ$ .

step 2.1A4A7algebra

4.1

For self-adjoint $T$ the self-adjoint calculus of [A5] is an isometric unital star-homomorphism $C (σ (T)) \to B (H)$ with $z \mapsto T$ and range $C^{*} (I, T)$ ; regarded as a map onto $C^{*} (I, T)$ it is an isometric unital $*$ -isomorphism, so step 3.1 identifies it with $Ψ$ .

step 2.1step 3.1A5A6

5.1

The map $f \mapsto f (T) : = Ψ (f)$ is therefore the unique isometric unital star-isomorphism $C (σ (T)) \to C^{*} (I, T)$ sending $z$ to $T$ , and it agrees with the self-adjoint calculus when $T$ is self-adjoint.

step 2.1step 3.1step 4.1∎

Depends on

Used by

Continuous functional calculus cannot produce every spectral projection Counterexample
Cyclic vector and cyclic normal operator Definition
Functional calculus for a diagonal operator Example
Spectral projection of an isolated eigenvalue agrees with the riesz projection Example
Continuous functional calculus produces a regular PVM Lemma
Borel functional calculus for bounded normal operators Theorem
Bounded normal operator abstract spectral theorem Theorem
Continuous functional calculus properties Theorem
Cyclic spectral representation Theorem
Multiplication operator form of the bounded normal spectral theorem Theorem
Spectral mapping for continuous normal functional calculus Theorem
Spectral theorem for bounded normal operators pvm form Theorem
Support and uniqueness of the spectral measure Theorem

Dependency tree · two levels

49 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 5.70, printed pp.268–273 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem, Example 4.9, pp.13–15 (standard reference, not scraped)