Journal of Operator Theory

Volume 79, Issue 2, Spring 2018 pp. 345-372.

Towards a perturbation theory for eventually positive semigroups

Authors: Daniel Daners (1) and Jochen Gluck (2)
Author institution:(1) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
(2) Institut fuer Angewandte Analysis, Universitaet Ulm, D-89069 Ulm, Germany

Summary: 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 perturbations 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.

DOI: http://dx.doi.org/10.7900/jot.2017mar29.2148
Keywords: one-parameter semigroups of linear operators, semigroups on Banach lattices, eventually positive semigroup, Perron--Frobenius theory, perturbation theory

Contents Full-Text PDF