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Individual stability of $$C_0$$-semigroups with uniformly bounded local resolvent. (English) Zbl 0892.47040
In the spirit of L. Gearhart’s theorem [Trans. Am. Math. Soc. 236, 385-394 (1978; Zbl 0371.47033)] the author proves the following individual stability theorem for strongly continuous semigroups $$(T(t))$$ on a Banach space X with generator A. If, for some $$x_0 \in X$$, the map $$z \mapsto R(z,A)x_0$$ has a bounded analytic extension to $$\{z:\text{Re}z > 0\}$$ then $$\| T(t)R(\lambda, A)x_0\| \leq M(1 + t)$$ for all $$t\geq 0$$, some (all) $$\lambda \in \varrho(A)$$ and some $$M \in \mathbb R_+$$. As a corollary he obtains the recent theorem by L. Weis and V. Wrobel [Proc. Am. Math. Soc. 124, No. 12, 3663-3671 (1996; Zbl 0863.47027)].

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations
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##### References:
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