## 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
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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.## References

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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