Continuous semigroup

Question

Let $X$ Banach, suppose that $A: D(A) \subset X \to X$ is the infinitesimal generator of a strongly continuous semigroup. $e^{At}, t \geq 0$. Then is the following number finite?

$$M=\sup_{0 \leq t \leq T} |{e^{At}}|_{L(X)}$$

Answer 1

Yes, it is. First note that there exists $\delta>0$ such that $\sup_{t\leq \delta}\|e^{tA}\|<\infty$. For if not, there would be a sequence null sequence $(\delta_n)$ such that $\|e^{\delta_n A}\|\to\infty$. By the uniform boundedness principle, this implies the existence of $x\in X$ such that $\|e^{\delta_n A}x\|$ is unbounded, contradicting the strong continuity of $(e^{tA})$.

Now if $T>0$ is arbitrary, there exists $N\in \mathbb{N}$ such that $T/N\leq \delta$, and for $t\in [0,T]$ we have $$ \sup_{t\leq T}\|e^{tA}\|=\sup_{t\leq T}\|(e^{t/N A})^N\|\leq \sup_{s\leq \delta}\|e^{sA}\|^N<\infty. $$

Answer 2

It is obvious from the exponential bound for $C_0$-semigroups: there exist always $M\ge 1$ and $\omega \in \mathbb{R}$ such that $$\|e^{tA}\| \leq M e^{\omega t} \quad \text{for all } t\ge 0.$$ The proof is basically the same as in the answer by @MaoWao.