Strongly continuous one-parameter unitary group

A strongly continuous one-parameter unitary group on the complex Hilbert space $H$ is a map $U:\mathbb{R}\to \mathcal{B}(H)$ such that $U(0)=I$, $U(s+t)=U(s)U(t)$ for all $s,t\in \mathbb{R}$, each $U(t)$ is unitary (that is, onto and norm preserving, equivalently $U(t{)}^{\ast }U(t)=U(t)U(t{)}^{\ast }=I$, A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum), and the orbit maps $t\mapsto U(t)x$ are continuous at every $t$ for every $x\in H$, from the usual metric on $\mathbb{R}$ to the norm metric of $H$ (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form).

Continuity at the origin suffices, and weak continuity is equivalent to strong continuity. These two clauses are part of the definition's content and are proved here. If $t\mapsto U(t)x$ is continuous at $t=0$ and $t_0\in \mathbb{R}$, then $U(t)x-U(t_0)x=U(t_0)(U(t-t_0)x-x)$ and $U(t_0)$ is isometric, so $\mathrm{\parallel }U(t)x-U(t_0)x\mathrm{\parallel }=\mathrm{\parallel }U(t-t_0)x-x\mathrm{\parallel }\to 0$ as $t\to t_0$; the group law and isometry turn continuity at one point into continuity everywhere. Likewise, if $t\mapsto U(t)x$ is merely weakly continuous at $0$, then for $x\in H$ the expansion $\mathrm{\parallel }U(t)x-x{\mathrm{\parallel }}^2=2\mathrm{\parallel }x{\mathrm{\parallel }}^2-2\mathrm{Re}⟨U(t)x,x⟩$ shows that norm convergence at $t=0$ follows from $⟨U(t)x,x⟩\to ⟨x,x⟩$; and at an arbitrary $t_0$ one has $U(t)y-U(t_0)y=U(t_0)(U(t-t_0)y-y)$, so weak continuity at $t_0$ for every $y$ follows from the case $t=0$. Conversely, norm continuity of an orbit implies its weak continuity, since for each fixed $y\in H$, Cauchy--Schwarz gives $\mathrm{\mid }⟨U(t)x-U(t_0)x,y⟩\mathrm{\mid }\le \mathrm{\parallel }U(t)x-U(t_0)x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }⟶0.$ Finally, a group with $U(0)=I$ is automatically invertible with $U(t{)}^{-1}=U(-t)$, so the unitarity and group clauses are symmetric in $t$.