I have a question about analytic semigroups. I know that if $A: D(A)\subset X\to X$ is a sectorial operator on $X$, it generates an analytic semigroup $e^{tA}$. One of the properties of this semigroup is the following: there exists constant $M_0$ such that $\left\| e^{tA}\right\|_{L(X)}\leq M_0 e^{\omega t}$, $t>0$. What can I say about $\left\| A e^{tA}\right\|_{L(X)}$? Can I also find some constant $\hat{M}$, such that $\left\| A e^{tA}\right\|_{L(X)}\leq \hat{M} e^{\omega t} $ ?
In the literature I have seen the estimation of type $\left\|t A e^{tA}\right\|_{L(X)}\leq C_{\varepsilon} e^{(\omega + \varepsilon) t}$. But I need the estimation of $\left\| A e^{tA}\right\|_{L(X)}$.
I would really appreciate any help!