Let $\{T(t)\}_{t\geq 0}$ be a $C_{0}$-semigroup on a Banach space $X$ and define the growth bound to be the quantity \begin{equation} \omega_{A}=\inf\{\omega\in\mathbb{R} \mid \exists\text{ } M_{\omega}\geq 1 \text{ s.t. } \|T(t)\|_{X\to X}\leq M_{\omega} e^{\omega t} \text{ }\forall\text{ }t\geq 0\} \end{equation} where $\omega_{A}=-\infty$ is allowable. Suppose now that $\omega_{A}\leq 0$. Clearly, if $\omega_{A}<0$, then there exists $\epsilon>0$ such that \begin{equation} \|T(t)\|_{X\to X}\leq M_{-\epsilon}e^{-\epsilon t}\leq M_{-\epsilon}\end{equation} so that the semigroup decays exponentially and is therefore uniformly bounded as well. The semigroup need not even be uniformly bounded if $\omega_{A}=0$, however, it follows that for every $\epsilon>0$, there exists $M_{\epsilon}\geq 1$ such that \begin{equation} \|T(t)\|_{X\to X}\leq M_{\epsilon} e^{\epsilon t} \end{equation} and my question is then the following: can anything in general be said here about the growth of the constants $M_{\epsilon}$ as $\epsilon\to 0^{+}$? In particular, can we say something like $M_{\epsilon}\leq C\epsilon^{-n}$ for some fixed constant $C\geq 1$ and positive integer $n$?
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