On norm continuity, differentiability and compactness of perturbed semigroups

TL;DR

This paper investigates how perturbations affect properties like norm continuity, compactness, and differentiability of semigroups in Banach spaces, using feedback theory and applying results to boundary integro-differential equations.

Contribution

It provides new insights into the effects of Miyadera-Voigt, Desch-Schappacher, and Staffans-Weiss perturbations on semigroup properties in Banach spaces.

Findings

Established conditions for norm continuity under perturbations

Analyzed compactness and differentiability preservation

Applied results to boundary integro-differential equations

Abstract

The main purpose of this paper is to treat semigroups properties, like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations, Miyadera-Voigt, Desch-Schappacher and Staffans-Weiss perturbations. Our approach is mainly based on feedback theory of Salamon-Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces.