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

DOI:
https://doi.org/10.1090/S0002-9939-1989-0953013-0

MathSciNet review:
953013

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Abstract: Let , be a strongly continuous semigroup on a Hilbert space (or, more generally, on a reflexive Banach space with the approximating property), with infinitesimal generator . Let: (i) either or be strongly stable, yet (ii) not uniformly stable as . Then, for any compact operator on , the semigroup generated by cannot be uniformly stable as . This result is 'optimal' within the class of compact perturbations . It improves upon a prior result in [G.1] by removing the assumption that be a contraction for positive times. Moreover, it complements a result in [R.1] where 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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DOI:
https://doi.org/10.1090/S0002-9939-1989-0953013-0

Article copyright:
© Copyright 1989
American Mathematical Society