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Continuous functional calculus properties

Statement

Assume AC. The continuous functional calculus on a nonzero complex Hilbert space preserves sums, products, conjugation and positivity, is norm-continuous, obeys continuous composition, sends eigenvectors at $λ$ to scalar evaluation at $λ$ , and commutes with every $S$ satisfying $S T = T S$ and $S T^{*} = T^{*} S$ .

Facts & Assumptions

[A1]

The normal calculus $f \mapsto f (T)$ is an isometric unital star-isomorphism $C (σ (T)) \to C^{*} (I, T)$ , hence complex-linear, multiplicative, unital, star-preserving and isometric; when $T$ is self-adjoint, this normal calculus agrees function-by-function with the self-adjoint calculus, which has the same properties and range (Continuous functional calculus for bounded normal operators, Continuous functional calculus for bounded self adjoint operators).

[A2]

$σ (f (T)) = f (σ (T))$ and $f (T)$ is normal (Spectral mapping for continuous normal functional calculus).

[A3]

$*$ -polynomials are uniformly dense in the continuous functions on a compact subset of $C$ (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense).

[A4]

$C^{*} (I, T)$ is the norm closure of the unital $*$ -algebra of $*$ -polynomials in $T$ , and multiplication in $B (H)$ is continuous (C star algebra generated by a normal operator, Hilbert-adjoint identities).

[A5]

For normal $T$ and $T x = λ x$ one has $T^{*} x = \overline{λ} x$ : $∥ (T^{*} - \overline{λ}) x ∥^{2} = ⟨ (T^{*} - \overline{λ}) (T - λ) x, x ⟩ = ⟨ (T - λ) x, (T - λ) x ⟩ = 0$ by normality (Self-adjoint, positive, unitary and normal operators, Hilbert-adjoint identities).

[A6]

If $X \geq 0$ is a bounded positive operator then $σ (X) \subseteq [0, + \infty)$ (Spectrum of a positive operator is nonnegative).

[A7]

AC is the hypothesis of the calculus and spectral suppliers (The Axiom of Choice).

Proof

technique · direct

Given: A nonzero complex Hilbert space $H$ , a bounded normal $T \in B (H)$ , functions $f, k \in C (σ (T))$ , a function $h \in C (σ (f (T)))$ , and an operator $S$ commuting with $T$ and $T^{*}$ .

1.1

Linearity, multiplicativity, conjugation and norm-continuity hold: $f (T)$ is the image of $f$ under an isometric unital $*$ -isomorphism, so $(f + k) (T) = f (T) + k (T)$ , $(f k) (T) = f (T) k (T)$ , $(λ f) (T) = λ f (T)$ , $\overline{f} (T) = f (T)^{*}$ and $∥ f (T) ∥ = ∥ f ∥_{\infty}$ .

1.2

If $f \geq 0$ is real-valued, write $f = ∣ r ∣^{2}$ with $r = \sqrt{f}$ continuous; then $f (T) = r (T)^{*} r (T)$ and $⟨ f (T) x, x ⟩ = ∥ r (T) x ∥^{2} \geq 0$ , so $f (T)$ is a positive operator; conversely, if $f (T) \geq 0$ as a quadratic form, then $σ (f (T)) \subseteq [0, + \infty)$ and [A2] gives $f (σ (T)) \subseteq [0, + \infty)$ , so $f \geq 0$ .

A1A2A6algebra

1.3

For every $*$ -polynomial $q$ in $z, \overline{z}$ and every eigenvector $x$ of $T$ at $λ$ , one has $T^{*} x = \overline{λ} x$ and hence $q (T, T^{*}) x = q (λ, \overline{λ}) x$ .

A4A5algebra

1.4

If $S$ commutes with $T$ and $T^{*}$ , then $S$ commutes with every $*$ -polynomial in $T$ , and hence with every element of $C^{*} (I, T)$ by continuity of multiplication.

A4algebra

2.1

Eigenvector identity: for eigenvectors $x$ of $T$ at $λ$ and $f$ continuous, approximate $f$ uniformly on $σ (T)$ by $*$ -polynomials $q_{n}$ ; then $f (T) x = \lim q_{n} (T) x = \lim q_{n} (λ, \overline{λ}) x = f (λ) x$ .

step 1.3A1A3

2.2

Composition: since $f (T)$ is normal and $σ (f (T)) = f (σ (T))$ , choose $*$ -polynomials $q_{n}$ converging uniformly to $h$ on $σ (f (T))$ . Multiplicativity and conjugation give $q_{n} (f (T), f (T)^{*}) = (q_{n} (f, \overline{f})) (T)$ . The left side converges to $h (f (T))$ by the isometry of the calculus for $f (T)$ , while the right side converges to $(h \circ f) (T)$ because $q_{n} \circ f \to h \circ f$ uniformly on $σ (T)$ . Thus $h (f (T)) = (h \circ f) (T)$ .

step 1.1A1A2A3algebra

2.3

Commutant: since $f (T) \in C^{*} (I, T)$ and $S$ commutes with all of $C^{*} (I, T)$ , $S f (T) = f (T) S$ .

step 1.4A4

3.1

The calculus therefore preserves sums, products, conjugation and positivity, is norm-continuous, obeys continuous composition $h (f (T)) = (h \circ f) (T)$ , sends eigenvectors at $λ$ to the scalar $f (λ)$ , and commutes with every $S$ commuting with $T$ and $T^{*}$ .

step 1.1step 1.2step 2.1step 2.2step 2.3A7∎

Depends on

Used by

Spectral projections and resolution of the identity Corollary
Borel functional calculus for a bounded normal operator Definition
Cyclic vector and cyclic normal operator Definition
Functional calculus for a diagonal operator Example
Sign and positive negative parts of a self adjoint operator Example
Continuous functional calculus produces a regular PVM Lemma
Borel functional calculus for bounded normal operators Theorem
Positive square root Theorem
Self adjoint norm and spectrum extrema Theorem
Spectral theorem for bounded normal operators pvm form Theorem

Dependency tree · two levels

44 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 5.54 and Theorem 5.70, printed pp.250–273 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem, Example 4.9, pp.13–15 (standard reference, not scraped)