Stone's theorem for one-parameter unitary groups

Statement

Let $U:\mathbb R\to U(\mathcal H)$ be a strongly continuous unitary representation of the additive group, equivalently a strongly continuous unitary one-parameter group. Stone's theorem states that there is a unique, possibly unbounded, self-adjoint operator $A$ on $\mathcal H$ such that

U(t)=e^{itA}\qquad(t\in\mathbb R).

Conversely, every self-adjoint $A$ defines such a group through the spectral functional calculus. Its domain is

\operatorname{Dom}(A)=\left\{\xi:\lim_{t\to0}\frac{U(t)\xi-\xi}{t}\text{ exists in norm}\right\},

and the displayed limit equals $iA\xi$ .

Generator and domain

The strong derivative at $0$ is therefore not defined on every vector unless $A$ is bounded. On its dense domain it gives the closed skew-adjoint operator $iA$ . The group determines both the operator and its domain: specifying only a formal derivative or an eigenvalue formula does not specify an unbounded generator.

Consequences

Strong continuity, the group law, and unitarity force differentiability precisely on $\operatorname{Dom}(A)$ , even though no differentiability is assumed initially. The spectral theorem then yields

U(t)=\int_{\mathbb R}e^{it\lambda}\,dE_A(\lambda).

In a unitary representation of a Lie group, applying Stone's theorem to $t\mapsto\pi(\exp(tX))$ produces the self-adjoint generator associated with each Lie-algebra element $X$ Reed–Simon, Theorem VIII.8.

Sign conventions

Some authors write $U(t)=e^{-itH}$ . Their self-adjoint generator $H$ is $-A$ , and the strong derivative is $-iH$ . Other authors call the skew-adjoint operator $iA$ , rather than $A$ , the infinitesimal generator. The exponential formula and derivative-domain identity must be stated together to remove this ambiguity.

References

M. H. Stone, “On One-Parameter Unitary Groups in Hilbert Space,” Annals of Mathematics 33 (1932), 643–648. DOI record. Relevant: the original correspondence theorem.
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. DOI record. Relevant: Theorem VIII.8.