Towards a perturbation theory for eventually positive semigroups

TL;DR

This paper investigates the stability of eventual positivity in operator semigroups under perturbations, revealing that it is generally sensitive to positive perturbations and providing conditions for preservation.

Contribution

It establishes that eventual positivity is not stable under large positive perturbations and offers criteria for when positive perturbations preserve eventual positivity.

Findings

Eventual positivity is not stable under large positive perturbations.

If eventual positivity is preserved under all positive perturbations, the semigroup must be positive.

Provided explicit bounds and conditions for positive perturbations to maintain eventual positivity.

Abstract

We consider eventually positive operator semigroups and study the question whether their eventual positivity is preserved by bounded perturbations of the generator or not. We demonstrate that eventual positivity is not stable with respect to large positive perturbation and that certain versions of eventual positivity react quite sensitively to small positive perturbations. In particular we show that if eventual positivity is preserved under arbitrary positive perturbations of the generator, then the semigroup is positive. We then provide sufficient conditions for a positive perturbation to preserve the eventual positivity. Some of these theorems are qualitative in nature while others are quantitative with explicit bounds.