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On individual stability of $C_0-$semigroups

Author: J. M. A. M. van Neerven
Journal: Proc. Amer. Math. Soc. 130 (2002), 2325-2333
MSC (2000): Primary 47D03; Secondary 47D06
Published electronically: February 4, 2002
MathSciNet review: 1896416
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Abstract: Let $\{T(t)\}_{t\ge 0}$ be a $C_0$-semigroup with generator $A$ on a Banach space $X$. Let $x_0\in X$ be a fixed element. We prove the following individual stability results.

(i) Suppose $X$ is an ordered Banach space with weakly normal closed cone $C$and assume there exists $t_0\ge 0$ such that $T(t)x_0\in C$ for all $t\ge t_0$. If the local resolvent $\lambda\mapsto ( \lambda-A)^{-1} x_0$ admits a bounded analytic extension to the right half-plane $\{\operatorname{Re}\lambda>0\}$, then for all $\mu\in\varrho(A)$ and $x^\ast\in X^\ast$ we have

\begin{displaymath}\lim_{t\to\infty} \bigl\langle T(t)(\mu-A)^{-1} x_0, x^\ast\bigr\rangle\, =\, 0.\end{displaymath}

(ii) Suppose $E$ is a rearrangement invariant Banach function space over $[0,\infty)$ with order continuous norm. If $x_0^\ast\in X^\ast$ is an element such that $t\mapsto \langle T(t)x_0, x_0^\ast\rangle$ defines an element of $E$, then for all $\mu\in\varrho(A)$ and $\beta\ge 1$ we have

\begin{displaymath}\lim_{t\to\infty} \bigl\langle T(t)(\mu-A)^{-\beta} x_0, x_0^\ast\bigr\rangle\, =\, 0.\end{displaymath}

References [Enhancements On Off] (What's this?)

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Additional Information

J. M. A. M. van Neerven
Affiliation: Department of Applied Mathematical Analysis, Technical University of Delft, P.O. Box 5031, 2600 GA Delft, The Netherlands

Keywords: Individual stability, bounded local resolvent, weakly normal cone, positive semigroup, $C_0-$semigroup, rearrangement invariant, Banach function space, order continuous norm
Received by editor(s): April 5, 2000
Received by editor(s) in revised form: March 1, 2001
Published electronically: February 4, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society