Eventually Positive Semigroups of Linear Operators

TL;DR

This paper develops a comprehensive theory of eventually positive semigroups of linear operators, characterizing their properties and applying these to various examples including elliptic operators and delay differential equations.

Contribution

It introduces a systematic framework for analyzing eventually positive semigroups, extending spectral results to non-positive cases on Banach lattices.

Findings

Characterizations via resolvent and spectral conditions.

Proven eventual positivity for specific elliptic and delay operators.

Extended spectral bound results to eventually positive semigroups.

Abstract

We develop a systematic theory of eventually positive semigroups of linear operators mainly on spaces of continuous functions. By eventually positive we mean that for every positive initial condition the solution to the corresponding Cauchy problem is positive for large enough time. Characterisations of such semigroups are given by means of resolvent properties of the generator and Perron--Frobenius type spectral conditions. We apply these characterisations to prove eventual positivity of several examples of semigroups including some generated by fourth order elliptic operators and a delay differential equation. We also consider eventually positive semigroups on arbitrary Banach lattices and establish several results for their spectral bound which were previously only known for positive semigroups.