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Lack of Uniform Stabilization for Noncontractive Semigroups Under Compact Perturbation
Proceedings of the American Mathematical Society
Vol. 105, No. 2 (Feb., 1989), pp. 375-383
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2046953
Page Count: 9
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Let $G(t), t \geq 0$, be a strongly continuous semigroup on a Hilbert space $X$ (or, more generally, on a reflexive Banach space with the approximating property), with infinitesimal generator $A$. Let: (i) either $G(t)$ or $G^\ast(t)$ be strongly stable, yet (ii) not uniformly stable as $t \rightarrow +\infty$. Then, for any compact operator $B$ on $X$, the semigroup $S_B(t)$ generated by $A + B$ cannot be uniformly stable as $t \rightarrow +\infty$. This result is `optimal' within the class of compact perturbations $B$. It improves upon a prior result in [G.1] by removing the assumption that $G(t)$ be a contraction for positive times. Moreover, it complements a result in [R.1] where $G(t)$ was assumed to be a group, contractive for negative times. Our proof is different from both [R.1 and G.1]. Application include physically significant dynamical systems of hyperbolic type in feedback form, where the results of either [R.1 or G.1] are not applicable, as the free dynamics is not a contraction.
Proceedings of the American Mathematical Society © 1989 American Mathematical Society