Let $\alpha:\mathbb R\to U(H)$ be a strongly continuous action of the reals on some Hilbert space, and let $A=-i\frac d{dt}\alpha(t)|\_{t=0}$ be its infinitesimal generator, so that $\alpha(t)=e^{itA}$. Finally, let $D\subset H$ be the domain of $A$.


A subset $D\_0\subset D$ is called a core of $A$ if the closure of $A|\_{D\_0}$ is $A$.


I've read the following statement in some paper, without reference or explanation, and I'd like to know why it's true:



> 
> If $D\_0$ is dense in $H$ and invariant under $\alpha$, then it is a core of $A$.
> 
> 
> 



why is this a sufficient condition for a domain to be a core of an unbounded operator?
This is stated and proved as Proposition 2.20 here:
<https://isem-mathematik.uibk.ac.at/isemwiki/images/9/94/ISEM15_Lecture2.pdf>


