Irreducibility of eventually positive semigroups

TL;DR

This paper extends the theory of irreducibility from positive semigroups to eventually positive semigroups, developing new tools to analyze their spectral properties and invariance structures.

Contribution

It introduces novel methods to characterize irreducibility and spectral behavior of eventually positive semigroups, which were previously inaccessible with classical positivity techniques.

Findings

Characterization of irreducibility via linear functionals

Perturbation results for eventually positive semigroups

Spectral theorems applicable to the eventually positive case

Abstract

Positive $C_0$-semigroups that occur in concrete applications are, more often than not, irreducible. Therefore a deep and extensive theory of irreducibility has been developed that includes characterizations, perturbation analysis, and spectral results. Many arguments from this theory, however, break down if the semigroup is only eventually positive - a property that has recently been shown to occur in numerous concrete evolution equations. In this article, we develop new tools that also work for the eventually positive case. The lack of positivity for small times makes it necessary to consider ideals that might only be invariant for large times. In contrast to their classical counterparts - the invariant ideals - such eventually invariant ideals require more involved methods from the theory of operator ranges. Using those methods we are able to characterize (an appropriate adaptation of) irreducibility by means of linear functionals, derive a perturbation result, prove a number of spectral theorems, and analyze the interaction of irreducibility with analyticity, all in the eventually positive case. By a number of examples, we illustrate what kind of behaviour can and cannot be expected in this setting.