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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Lack of uniform stabilization for noncontractive semigroups under compact perturbation
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by R. Triggiani
Proc. Amer. Math. Soc. 105 (1989), 375-383
DOI: https://doi.org/10.1090/S0002-9939-1989-0953013-0

Abstract:

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 ${G^ * }(t)$ be strongly stable, yet (ii) not uniformly stable as $t \to + \infty$. Then, for any compact operator $B$ on $X$, the semigroup ${S_B}(t)$ generated by $A + B$ cannot be uniformly stable as $t \to + \infty$. 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 dyamics is not a contraction.
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Bibliographic Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 105 (1989), 375-383
  • MSC: Primary 47D05
  • DOI: https://doi.org/10.1090/S0002-9939-1989-0953013-0
  • MathSciNet review: 953013