Let $A$ be a self-adjoint operator on a Hilbert space such that $A$ is the generator of a $C_0$-semigroup $(T_t)_{t\ge 0}$. Must $(T_t)_{t\ge 0}$ be analytic?
I know that self-adjoint operators on Hilbert space generate a bounded analytic semigroup if and only if they are sectorial. However, I was wondering if we're already given a self-adjoint operator is a generator, can we then guarantee that the semigroup would be analytic?
I haven't been able to locate a counter-example.