Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space) and let \begin{align*} A:\mathcal D(A)\subset H\to H \end{align*} be self-adjoint. Then the spectral calculus defines a strongly continuous unitary group $(U(t))_{t\in\mathbb R}$ by \begin{align*} U(t)=e^{-itA}. \end{align*} This group has self-adjoint generator $A$: for every $u\in\mathcal D(A)$, $U(t)u\in\mathcal D(A)$ and the curve $t\mapsto U(t)u$ solves \begin{align*} i\frac{d}{dt}U(t)u=A U(t)u \end{align*} with initial value $u$. Conversely, a strongly continuous unitary group has a unique self-adjoint generator in this sense.