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Strong continuity of positive modular unitary powers

Statement

Assume the Axiom of Choice. Let $H$ be a complex Hilbert space and let $A$ be a positive self-adjoint injective operator with spectral projection valued measure $E$ supported on $(0, \infty)$ . For $t \in R$ , the unbounded Borel functional calculus defines $A^{i t}$ as a unitary on all of $H$ . For every $ξ \in H$ , the map $t \mapsto A^{i t} ξ$ is norm-continuous.

Statement references

The Axiom of Choice

Integral of a measurable function against a projection-valued measure

Spectral theorem for unbounded self-adjoint operators (PVM form)

Unbounded Borel functional calculus: domains, products, spectral mapping

Dominated convergence

Facts & Assumptions

Given: The Axiom of Choice, a complex Hilbert space $H$ , a positive self-adjoint injective $A$ with spectral measure $E$ , and $ξ \in H$ .

[A1]

Assume the Axiom of Choice (The Axiom of Choice).

[F1]

The spectral theorem and unbounded Borel calculus give the scalar spectral measure $μ_{ξ} (B) = ⟨ E (B) ξ, ξ ⟩$ , the norm formula for bounded spectral multipliers, and unitary imaginary powers (Spectral theorem for unbounded self-adjoint operators (PVM form), Integral of a measurable function against a projection-valued measure, Unbounded Borel functional calculus: domains, products, spectral mapping).

[F2]

Dominated convergence applies to the finite scalar measure $μ_{ξ}$ (Dominated convergence).

Proof

1.1A1F1F2givenalgebra

Let $t_{n} \to t$ . For every $λ > 0$ , $λ^{i t_{n}} \to λ^{i t}$ , while $∣ λ^{i t_{n}} - λ^{i t} ∣^{2} \leq 4$ . Since $μ_{ξ} ((0, \infty)) = ∥ ξ ∥^{2} < \infty$ , [F1] and [F2] give $∥ A^{i t_{n}} ξ - A^{i t} ξ ∥^{2} = \int_{(0, \infty)} ∣ λ^{i t_{n}} - λ^{i t} ∣^{2} d μ_{ξ} (λ) ⟶ 0.$ Thus the orbit is sequentially continuous.

2.1step 1.1algebra∎

The real parameter space and Hilbert-space norm topology are metric, so sequential continuity is equivalent to continuity. Therefore $t \mapsto A^{i t} ξ$ is norm-continuous for every $ξ \in H$ .

Depends on

Dependency tree · 2 levels

Sources

F. Hiai, Concise lectures on selected topics of von Neumann algebras