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Stone's theorem: unitary groups and self-adjoint generators

Statement

Assume the Axiom of Choice. The map $T \mapsto U (t) = e^{i t T}$ , computed by the Borel functional calculus of $T$ (Unbounded Borel functional calculus: domains, products, spectral mapping), is a bijection from the set of self-adjoint operators on $H$ onto the set of strongly continuous one-parameter unitary groups on $H$ . Its inverse assigns to $U$ its generator $G$ and the self-adjoint operator $T = - i G$ ; here $D (T) = D (G)$ is exactly the set of vectors at which $t \mapsto U (t) x$ is differentiable at $0$ , and $\frac{1}{t} (U (t) x - x) \to i T x$ for $x \in D (T)$ .

Facts & Assumptions

[A1]

For every self-adjoint $T$ , $U (t) = e^{i t T}$ is a strongly continuous one-parameter unitary group with generator $G = i T$ and $D (G) = D (T)$ (A self-adjoint operator generates a strongly continuous unitary group).

[A2]

The generator $G$ of a strongly continuous unitary group is densely defined and skew-adjoint, so $S : = - i G$ is self-adjoint with $D (S) = D (G)$ (The generator of a unitary group is closed and skew-adjoint, Infinitesimal generator of a unitary group).

[A3]

If $x \in D (G)$ then $U (t) x \in D (G)$ , $G U (t) x = U (t) G x$ , and $t \mapsto U (t) x$ is differentiable with derivative $U (t) G x$ (Laplace resolvents of a unitary group, Infinitesimal generator of a unitary group).

[A4]

A skew-symmetric operator $G$ satisfies $Re ⟨ G w, w ⟩ = 0$ for $w \in D (G)$ , because $⟨ G w, w ⟩ = - ⟨ w, G w ⟩ = - \overline{⟨ G w, w ⟩}$ . The generator in [A2] is skew-adjoint and hence skew-symmetric. The generator of a unitary group is closed and skew-adjoint Hilbert space

Proof

technique · direct

Given: A self-adjoint $T$ , and a strongly continuous unitary group $U$ with generator $G$ .

1.1

Applying [A1] to $T$ produces a strongly continuous unitary group with generator $i T$ , so the map $T \mapsto e^{i t T}$ is well defined, and its derivative at $0$ exists exactly on $D (T)$ where it equals $i T x$ .

1.2

Applying [A2] to $U$ produces the self-adjoint operator $S = - i G$ with $D (S) = D (G)$ ; applying [A1] to $S$ gives the strongly continuous unitary group $V (t) = e^{i t S}$ , whose generator is $i S = G$ .

A1A2

2.1

Uniqueness for a fixed generator: if $U, V$ are strongly continuous unitary groups with the same generator $G$ and $x \in D (G)$ , then $w (t) : = U (t) x - V (t) x$ is differentiable with $w^{'} (t) = G w (t)$ by [A3], so $\frac{d}{d t} ∥ w (t) ∥^{2} = 2 Re ⟨ w (t), G w (t) ⟩ = 0$ by [A4], and $w (0) = 0$ gives $w \equiv 0$ on $D (G)$ ; since $D (G)$ is dense by [A2] and $U (t), V (t)$ are isometries, $U (t) = V (t)$ for every $t$ .

A2A3A4step 1.2

3.1

Hence $U = V$ in step 1.2, that is $U (t) = e^{i t T}$ for the self-adjoint $T = S = - i G$ ; combined with step 1.1 this makes $T \mapsto e^{i t T}$ a bijection with the stated inverse.

step 1.1step 1.2step 2.1

4.1

The derivative characterisation is the one from [A1] applied to the self-adjoint $T = - i G$ : the limit exists exactly on $D (T) = D (G)$ and equals $i T x$ .

A1step 3.1∎

Depends on

Used by

Periodic derivative and its unitary translation group Example

Dependency tree · two levels

58 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

Gerald Teschl, Mathematical Methods in Quantum Mechanics, second edition (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem (standard reference, not scraped)
Roland Schnaubelt, Evolution Equations (lecture notes) (standard reference, not scraped)