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

${\displaystyle (U_{t})_{t\in \mathbb {R} }}$

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

${\displaystyle \forall t_{0}\in \mathbb {R} ,\psi \in {\mathcal {H}}:\qquad \lim _{t\to t_{0}}U_{t}(\psi )=U_{t_{0}}(\psi ),}$

and are homomorphisms, i.e.,

${\displaystyle \forall s,t\in \mathbb {R} :\qquad U_{t+s}=U_{t}U_{s}.}$

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 John von Neumann (1932) showed that the requirement that ${\displaystyle (U_{t})_{t\in \mathbb {R} }}$ be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.

This is an impressive result, as it allows one to define the derivative of the mapping ${\displaystyle t\mapsto U_{t},}$ which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.