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

Question

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

Answer 1

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.

(See K.J. Engle, R . Nagel: One- Parameter Semigroups for Linear Evolution Equations, Theorem 4.6).

If $A$ is sectorial, then for the spectrum $ \sigma(A)$ we have

$$(*) \quad \sigma(A) \subset \{z \in \mathbb C: Re(z) \le 0\}.$$

(This is a consequence of the definition of "sectorial".)

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.