Infinitesimal generator of a unitary group

Let $U$ be a strongly continuous one-parameter unitary group on $H$ (Strongly continuous one-parameter unitary group). Its infinitesimal generator is the linear operator $G$ with domain $D(G):=\{x\in H:{\lim }_{t\to 0}\frac{1}{t}(U(t)x-x\left)\text{ exists in }H\right\},Gx:={\lim }_{t\to 0}\frac{1}{t}(U(t)x-x),$ where this is the two-sided norm limit over real $t\mathrm{\ne }0$: it has value $y$ precisely when for every $\epsilon >0$ there is $\delta >0$ such that $0<\mathrm{\mid }t\mathrm{\mid }<\delta ⟹\parallel \frac{U(t)x-x}{t}-y\parallel <\epsilon \mathrm{.}$ Equivalently, defining the quotient's value at $t=0$ to be $y$ makes it continuous there from the usual metric on $\mathbb{R}$ to the norm metric on $H$ (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form). Such a $y$ is unique by the triangle inequality.

$D(G)$ is a linear subspace and $G$ is linear: if $x,y\in D(G)$ and $a,b$ are scalars, then $\frac{1}{t}(U(t)(ax+by)-(ax+by))=a\frac{1}{t}(U(t)x-x)+b\frac{1}{t}(U(t)y-y)$ converges with limit $aGx+bGy$, because the operations of $H$ are continuous and both estimates use the same punctured real parameter $t$; the restriction to $D(G)$ is therefore well defined and linear (Linear subspace of a vector space, Linear map between vector spaces over the same field).

Sign convention. This page writes $T=-iG$ for the generator, so that Stone's theorem reads $U(t)=e^{itT}$ with $T$ self-adjoint; equivalently $G=iT$. In the convention of Teschl's book one has $U(t)=e^{-itA}$ with $A$ self-adjoint, so $G=-iA$ and hence $A=iG=-T$; every formula below is written in the $U(t)=e^{itT}$ convention and the translation is recorded where a source is cited.