= Stone's theorem on one-parameter unitary groups
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{title2=$U_t=e^{itA},\qquad A=A^*$}
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= Stone theorem for one-parameter unitary groups
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Every <strongly continuous unitary group> has a unique possibly unbounded <self-adjoint operator> $A$ as its generator. Its domain consists of the <vectors> for which the <norm> limit of $(U_t\xi-\xi)/(it)$ exists as $t\to0$. Conversely, the <Borel functional calculus for a normal operator> constructs a <strongly continuous unitary group> from every <self-adjoint> $A$. Smooth averaging supplies a dense generator domain; Laplace integrals of $U_t$ and $U_{-t}$ prove that $A\pm i$ are surjective, which establishes self-adjointness.
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