Generalized evolution semigroups and general dichotomies

TL;DR

This paper introduces a new class of evolution semigroups based on generalized semiflows, providing spectral characterizations that apply to various types of dichotomies, including those with time-varying rates.

Contribution

It develops a generalized framework for evolution semigroups and spectral analysis, extending applicability to non-exponentially bounded evolution families and diverse dichotomy types.

Findings

Spectral characterizations of generators for the new semigroups.

Application to a broad class of dichotomies, including time-varying ones.

Extension beyond exponentially bounded evolution families.

Abstract

We introduce a special class of real semiflows, which is used to define a general type of evolution semigroups, associated to not necessarily exponentially bounded evolution families. Giving spectral characterizations of the corresponding generators, our results directly apply to a wide class of dichotomies, such as those with time-varying rate of change.