Stone's theorem on one-parameter unitary groups

Remarks

The converse is a direct consequence of the spectral theorem and dominated convergence. For the main direction, $A$ is defined by the limit in the statement. Vectors $\int \varphi(t) U(t)x \, dt$ with smooth compactly supported $\varphi$ lie in its domain, so $A$ is densely defined. The group law shows that $A$ is symmetric, and the vectors $\int_0^\infty e^{-t} U(\pm t)x \, dt$ show that $A \pm iI$ are onto, which makes $A$ self-adjoint.

Example: on $L^2(\R)$, the translations $(U(t)f)(s) = f(s + t)$ form such a group. Its generator is $A = -i \, d/ds$ with domain the Sobolev space $H^1(\R)$, the momentum operator of quantum mechanics.

Strong continuity in clause 2 is needed only in a weak form: on a separable Hilbert space, weak measurability of $t \mapsto \ip{U(t)x}{y}$ already implies it, by a theorem of von Neumann.

Going deeper

For a student

The theorem is the unbounded counterpart of the fact that a continuous group of unitary matrices is $e^{itA}$ for a Hermitian matrix $A$. It gives a bijection between self-adjoint operators and strongly continuous unitary groups. It is the special case of semigroup theory where the semigroup extends to negative times and each operator is unitary; compare the Lumer–Phillips theorem. It is the starting point of the representation theory of $\R$, and with the SNAG theorem, of locally compact abelian groups.