PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.Existence.StoneGenerator

Let $H$ be a complex Hilbert space and let $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. This means $U(0) = I$, each $U(t)$ is a unitary operator, and for every vector $\xi \in H$, the map $t \mapsto U(t)\xi$ is continuous. Let $A$ be the candidate Stone generator associated with $U$, which is a partially defined linear operator mapping $\psi$ to $-i \left. \frac{d}{dt} U(t)\psi \right|_{t=0}$ for all $\psi$ in the domain where this derivative exists. A vector $x \in H$ is called an analytic vector for $A$ if $x$ is in the domain of $A^n$ for all $n \in \mathbb{N}$ and there exists some $t > 0$ such that the series $\sum_{n=0}^{\infty} \frac{\|A^n x\|}{n!} t^n$ converges. The theorem states that the set of analytic vectors for the candidate generator $A$ spans a dense subspace of $H$.

Let $H$ be a complex Hilbert space. Suppose $U: \mathbb{R} \to \mathcal{B}(H)$ is a strongly continuous one-parameter unitary group, satisfying $U(0) = I$, $U(t)$ is a unitary operator for each $t \in \mathbb{R}$, and the map $t \mapsto U(t)\xi$ is continuous for every $\xi \in H$. Then the candidate Stone generator $A$ of the group $U$, defined as $A\psi = -i \left. \frac{d}{dt} U(t)\psi \right|_{t=0}$ for all $\psi$ in the domain where the limit exists, is an essentially self-adjoint operator.