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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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