Self-adjoint operators and analytic semigroups

Question

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.

Answer 1

Yes. Indeed, a self-adjoint operator $A$ generates a $C_0$-semigroup if and only if it is bounded above, i.e there exists a constant $w\in \mathbb{R}$ s.t. $\langle Ax,x \rangle \le w \|x\|^2$ for all $x\in D(A)$. See e.g., Proposition 3.28, p. 91 in [1]. In this case, the generated semigroup is analytic of angle $\pi/2$. See e.g Corollary 4.7 and p. 106 in [1].

[1] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer-Verlag, 1999.