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 $A$.