Spectral properties of locally eventually positive operator semigroups

TL;DR

This paper studies the spectral properties and long-term behavior of locally eventually positive operator semigroups on Banach lattices, challenging existing assumptions and providing new necessary conditions for their spectra.

Contribution

It introduces minimal assumptions for locally eventually positive semigroups, showing that typical spectral conditions are not necessary, and explores their asymptotic behavior.

Findings

Typical spectral assumptions are not necessary for local eventual positivity

Provides necessary conditions on the peripheral point spectrum

Establishes results on the asymptotic behavior of orbits

Abstract

This paper considers strongly continuous semigroups of operators on Banach lattices which are locally eventually positive, a property that was first investigated in the context of concrete fourth-order evolution equations. We construct a simple example to show that the typical assumptions on the spectrum of the semigroup generator considered currently in the literature are far from necessary in the more general setting of local eventual positivity. Under minimal additional assumptions, we obtain results on the asymptotic behaviour of orbits, as well as necessary conditions on the peripheral point spectrum of locally eventually positive semigroups.