Stone's theorem on one-parameter unitary groups

In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space ${\displaystyle {\mathcal {H}}}$ and one-parameter families

of unitary operators that are strongly continuous, i.e.,

and are homomorphisms, i.e.,

Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

The theorem was proved by Marshall Stone (1930, 1932), and Von Neumann (1932) showed that the requirement that ${\displaystyle (U_{t})_{t\in \mathbf {R} }}$ be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.

...
Wikipedia