Robustness of strong stability of semigroups

TL;DR

This paper investigates conditions under which the strong stability of semigroups on Hilbert spaces is preserved under finite rank perturbations, extending existing results on polynomial stability.

Contribution

It characterizes classes of finite rank perturbations that maintain strong stability and improves results on polynomial stability preservation.

Findings

Finite rank perturbations can preserve strong stability under certain spectral conditions.

Polynomial boundedness of the resolvent near spectral points is crucial for stability preservation.

An example with a multiplication semigroup illustrates the theoretical results.

Abstract

In this paper we study the preservation of strong stability of strongly continuous semigroups on Hilbert spaces. In particular, we study a situation where the generator of the semigroup has a finite number of spectral points on the imaginary axis and the norm of its resolvent operator is polynomially bounded near these points. We characterize classes of finite rank perturbations preserving the strong stability of the semigroup. In addition, we improve recent results on preservation of polynomial stability of a semigroup under finite rank perturbations of its generator. Theoretic results are illustrated with an example where we consider the preservation of the strong stability of a multiplication semigroup.