I'm trying to understand the proof of Theorem 2.5.6 (chapter 2.5) in Amnon Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.

For the direction (a) $\implies$ (b) it would be helpful to have a proposition along the lines of:

An analytic semigroup of bounded linear operators on a sector $\Delta_\delta$ (see definition below) that is uniformly bounded on the non-negative real axis is uniformly bounded on every closed subsector of $\Delta_\delta$.

Does anyone know of a proof of this proposition? Or do you know a counterexample?

Thank you very much in advance!

For an analytic semigroup, I use the following definition:

Let $\Delta_\delta := \{ z \in \mathbb{C} : | \arg(z) | < \delta \}$. For $z \in \Delta_\delta$ let $T(z)$ be a bounded linear operator. The family $\{T(z)\}_{z \in \Delta_\delta}$ is an analytic semigroup on $\Delta_\delta$, if

• $z \to T(z)$ is analytic on $\Delta_\delta$.
• $T(0) = I$ and $T(z_1 + z_2) = T(z_1) T(z_2)$ for all $z_1, z_2 \in \Delta_\delta$.
• $\lim_{\substack{z \to 0 \\z \in \Delta_{\delta'}}} T(z)x = x$ for all $x \in X$ and $0< \delta' < \delta$.