If a $C^0$-semigroup on a Hilbert space has a dissipative self-adjoint generator, it is immediately differentiable

Question

Let $H$ be a $\mathbb R$-Hilbert space and $-A$ be the generator of a strongly continuous semigroup $(T(t))_{t\ge0}$ on $H$.

I've read that it would be a consequence of a suitable version of the spectral theorem that if $A$ is dissipative and self-adjoint, then $(T(t))_{t\ge0}$ is immediately differentiable$^1$. How can we prove that?

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$^1$ i.e. $T(t)x\in\mathcal D(A)$ for all $x\in H$ and $t>0$.

Answer 1

You have even a stronger result. A self-adjoint dissipative operator generates an analytic semigroup, which is, in particular, immediately differentiable.