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
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