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Lack of uniform stabilization for noncontractive semigroups under compact perturbation

Author: R. Triggiani
Journal: Proc. Amer. Math. Soc. 105 (1989), 375-383
MSC: Primary 47D05
MathSciNet review: 953013
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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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