Engel Nagel A Short Course on Operator Semigroups Corollary II.4.8 states:
(There should be a typo. If $\delta=0$ then the spectum is empty, but normal operator has a non-empty spectrum? Anyways,)
Then they proceed,
In particular, Corollary 4.8 shows that the semigroup generated by a self-adjoint operator A that is bounded above, which means that there exists $w\in\mathbb{R}$ such that $$ ( A x | x ) \leq w \| x \| ^ { 2 } \quad \text { for all } x \in D ( A ) $$ is analytic of angle $\pi/2$. Moreover, this semigroup is bounded if and only if $w\leq 0$.
I do not see how Corollary 4.8 implies this. If $w=0$, then $(Ax|x)\leq 0$, so it is still possible that $0\in \sigma(A)$. But $0\not\in \{z\in\mathbb{C}\mid |\arg(-z)|<\delta\}$.
Are we meant to consider something like $A+w$, consider ${e}^{t(A+w)}$ and then say ${e}^{tA}$ is well-defined further ${e}^{t(A+w)}={e}^{tA}e^{tw}$ etc.?