Let $A$ be the infinitesimal generator of analytic semigroup $S(t)$ on a Hilbert space such that: $$\|S(t)\|\le \frac{M}{t^{\gamma}}$$ what we get fot $$\|AS(t)\|\le ??$$
I really appreciate any help you can provide.
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Is there any other information on the resolvent of $A$? For example, does $0$ belong to $\rho(A)$? – Pedro Aug 29 '17 at 22:37
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yeah sure $0\in \rho(A)$, and the $\gamma$ is negative.. – Soufiane Aug 30 '17 at 09:53
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If $\gamma<0$, then $\frac{M}{t^\gamma}\to0$ as $t\to0$, so $S(t)\to0$ in the norm topology. This is clearly not the case. – Jason Aug 31 '17 at 21:54