Eventually positive semigroups: spectral and asymptotic analysis

TL;DR

This paper explores the spectral and asymptotic behavior of eventually positive semigroups on Banach lattices, extending existing theories and employing ultrapower techniques to analyze their properties.

Contribution

It advances the theory of eventual positivity by examining spectral cyclicity and asymptotics, especially in the irreducible case, using novel ultrapower methods.

Findings

Extended spectral cyclicity results for eventually positive semigroups.

Identified asymptotic trends in the irreducible case.

Developed ultrapower techniques to handle non-positivity arguments.

Abstract

The spectral theory of semigroup generators is a crucial tool for analysing the asymptotic properties of operator semigroups. Typically, Tauberian theorems, such as the ABLV theorem, demand extensive information about the spectrum to derive convergence results. However, the scenario is significantly simplified for positive semigroups on Banach lattices. This observation extends to the broader class of eventually positive semigroups -- a phenomenon observed in various concrete differential equations. In this paper, we investigate the spectral and asymptotic properties of eventually positive semigroups, focusing particularly on the persistently irreducible case. Our findings expand upon the existing theory of eventual positivity, offering new insights into the cyclicity of the peripheral spectrum and asymptotic trends. Notably, several arguments for positive operators and semigroups do not apply in our context, necessitating the use of ultrapower arguments to circumvent these challenges.