# Infinite-time admissibility under compact perturbations

## TL;DR

This paper explores how infinite-time admissibility of semigroups is affected by compact perturbations, demonstrating through examples that it is not necessarily preserved even under strong stability.

## Contribution

It provides the first examples showing that infinite-time admissibility can be lost under compact perturbations of the generator.

## Key findings

- Infinite-time admissibility is not preserved under compact perturbations.
- Both original and perturbed semigroups can be strongly stable.
- Examples illustrate the non-preservation of admissibility.

## Abstract

We investigate the behavior of infinite-time admissibility under compact perturbations. We show, by means of two completely different examples, that infinite-time admissibility is not preserved under compact perturbations $Q$ of the underlying semigroup generator $A$, even if $A$ and $A+Q$ both generate strongly stable semigroups.