Spectrum and convergence of eventually positive operator semigroups

Sahiba Arora, Jochen Glück

Published 2020-11-09Version 1

One of the many intriguing features of positive $C_0$-semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups. More specifically, the question whether the semigroup converges (if necessary, after a rescaling) as time tends to infinity can be characterized by a set of simple spectral and compactness conditions. Results of this type are known both for strong convergence and for convergence in the operator norm. In the present paper, we show that similar theorems remain true for the much larger class of (uniformly) eventually positive semigroups -- a class which has recently been shown to arise in the study of various concrete differential equations. A major step in our characterization of the operator norm convergence of eventually positive semigroups is to show that a version of the famous Niiro--Sawashima theorem holds not only for positive but also for eventually positive operators. Several proofs for positive operators and semigroups do not work in our setting any longer, which makes different arguments necessary and gives our approach a distinct flavour.