# Do self adjoint operators on a Hilbert space generates an analytic semigroup?

Is this generally true that, a densely defined, closed and self adjoint operator on a Hilbert space generates an analytic semigroup?

Let $$A$$ be a densely defined, closed and self adjoint operator on a Hilbert space. Then $$A$$ generates a bounded analytic semigroup $$\iff$$ $$A$$ is sectorial.
If $$A$$ is sectorial, then for the spectrum $$\sigma(A)$$ we have
$$(*) \quad \sigma(A) \subset \{z \in \mathbb C: Re(z) \le 0\}.$$
Now take a densely defined, closed and self adjoint operator $$A$$ such that $$(*)$$ does not hold. Then $$A$$ does not generate a bounded analytic semigroup.