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Borel functional calculus for bounded normal operators

Statement

Assume AC. Let $T$ be a bounded normal operator on a nonzero complex Hilbert space $H$ , let $E$ be its spectral projection valued measure on $σ (T)$ , and let $f \mapsto f (T)$ be the bounded Borel functional calculus of Borel functional calculus for a bounded normal operator. Then:

$f (T)$ extends the continuous calculus: for continuous $f$ it agrees with the operator denoted $f (T)$ by Continuous functional calculus for bounded normal operators, and $1_{B} (T) = E (B)$ for every Borel $B \subseteq σ (T)$ ;
the calculus is linear, unital, multiplicative and star-preserving: $(a f + b g) (T) = a f (T) + b g (T)$ , $1 (T) = I$ , $(f g) (T) = f (T) g (T)$ and $\overline{f} (T) = f (T)^{*}$ ;
$∥ f (T) x ∥^{2} = \int ∣ f ∣^{2} d E_{x}$ for every $x \in H$ and $∥ f (T) ∥ = ∥ f ∥_{E, \infty} \leq ∥ f ∥_{\infty}$ , the exact norm being the $E$ -essential supremum of $f$ ; in particular $f (T)$ is normal with $f (T)^{*} = \overline{f} (T)$ ;
if $(f_{n})$ are uniformly bounded Borel functions, $f$ is bounded Borel, and $f_{n} \to f$ pointwise $E$ -almost everywhere, then $f_{n} (T) \to f (T)$ in the strong operator topology;
every $S \in B (H)$ commuting with $T$ and $T^{*}$ commutes with every $f (T)$ .

Facts & Assumptions

[A1]

The spectral PVM $E$ of $T$ is the unique regular PVM on the Borel $σ$ -algebra of $σ (T)$ with $\int z d E = T$ ; it is obtained from the continuous calculus $π (f) = f (T)$ by the construction that represents each continuous functional $f \mapsto ⟨ π (f) x, y ⟩$ by a unique finite regular complex measure $μ_{x, y}$ with $\int f d μ_{x, y} = ⟨ f (T) x, y ⟩$ , and then $⟨ E (B) x, y ⟩ = μ_{x, y} (B)$ and $⟨ Φ_{E} (h) x, y ⟩ = \int h d μ_{x, y}$ for every bounded Borel $h$ (Continuous functional calculus produces a regular PVM, Spectral theorem for bounded normal operators pvm form).

[A2]

For continuous $f$ the calculus satisfies $Φ_{E} (f) = f (T)$ , and the map $f \mapsto f (T)$ on $C (σ (T))$ is a unital star-homomorphism that commutes with every $S$ satisfying $S T = T S$ and $S T^{*} = T^{*} S$ (Continuous functional calculus for bounded normal operators, Continuous functional calculus properties, Borel functional calculus for a bounded normal operator).

[A3]

$Φ_{E}$ is linear, unital, multiplicative and star-preserving on bounded Borel functions, $∥ Φ_{E} (f) ∥ = ∥ f ∥_{E, \infty}$ , $∥ Φ_{E} (f) x ∥^{2} = \int ∣ f ∣^{2} d E_{x}$ , and uniformly bounded pointwise $E$ -almost everywhere convergence implies strong convergence; by definition $f (T) = Φ_{E} (f)$ (Pvm integral is a star homomorphism, Bounded borel pvm integral, Borel functional calculus for a bounded normal operator).

[A4]

An operator $S$ commutes with $T$ and $T^{*}$ ; the adjoint satisfies $⟨ S x, y ⟩ = ⟨ x, S^{*} y ⟩$ and normal means $N N^{*} = N^{*} N$ (Hilbert-adjoint identities, Self-adjoint, positive, unitary and normal operators, Hilbert space).

[A5]

If two finite regular complex measures on a compact metric space have equal integrals against every continuous function then they are equal, because bounded complex functionals on $C (K; C)$ have a unique representing regular complex measure (The bounded complex dual of C_0(X) is regular complex measures).

[A6]

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 $H$ , a bounded normal operator $T$ with spectral PVM $E$ and Borel calculus $f \mapsto f (T) = Φ_{E} (f)$ , and a bounded operator $S$ commuting with $T$ and $T^{*}$ .

1.1

Clauses 1 to 4: the agreement with the continuous calculus and $1_{B} (T) = E (B)$ are the definition of the Borel calculus and the identification $Φ_{E} (1_{B}) = E (B)$ ; linearity, unitality, multiplicativity and conjugation preservation, the quadratic identity, the norm formula and the strong-convergence property are the corresponding properties of $Φ_{E}$ ; normality follows because $f (T) f (T)^{*} = Φ_{E} (f \overline{f}) = Φ_{E} (\overline{f} f) = f (T)^{*} f (T)$ .

A2A3

1.2

The measures $μ_{S x, y}$ and $μ_{x, S^{*} y}$ coincide: for every continuous $f$ one has $\int f d μ_{S x, y} = ⟨ f (T) S x, y ⟩ = ⟨ S f (T) x, y ⟩ = ⟨ f (T) x, S^{*} y ⟩ = \int f d μ_{x, S^{*} y}$ , using the commutant clause of the continuous calculus and the adjoint identity; both are finite regular complex measures, so equality of their integrals against all continuous functions gives $μ_{S x, y} = μ_{x, S^{*} y}$ .

A1A2A4

2.1

The commutant clause: for every bounded Borel $h$ and all $x, y \in H$ one has $⟨ h (T) S x, y ⟩ = \int h d μ_{S x, y} = \int h d μ_{x, S^{*} y} = ⟨ h (T) x, S^{*} y ⟩ = ⟨ S h (T) x, y ⟩$ , so $S h (T) = h (T) S$ ; in particular $S$ commutes with every spectral projection $E (B) = 1_{B} (T)$ and with every Borel calculus operator.

A1step 1.2A3A4A5

3.1

All five clauses hold: the Borel calculus extends the continuous calculus, is a unital star-homomorphism with $E$ -essential-supremum norm, is strongly continuous under uniformly bounded pointwise $E$ -almost everywhere convergence, and every operator commuting with $T$ and $T^{*}$ commutes with all of it.

step 1.1step 2.1A6∎

Depends on

Used by

Spectral projections and resolution of the identity Corollary
A normal operator need not have any eigenvectors Counterexample
Continuous functional calculus cannot produce every spectral projection Counterexample
Borel functional calculus defines a discontinuous characteristic function Example
Pvm of a diagonal normal operator Example
Pvm of a multiplication operator Example
Sign and positive negative parts of a self adjoint operator Example
Spectral projection of an isolated eigenvalue agrees with the riesz projection Example
Unitary intertwiners preserve direct-integral fiber dimension Lemma
Cyclic spectral representation Theorem
Stone resolvent formula for spectral projections Theorem
Support and uniqueness of the spectral measure Theorem

Dependency tree · two levels

62 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 5.75, printed pp.291–295 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem, Theorem 5.6(c), p.18 (standard reference, not scraped)