Eventually positive semigroups: spectral and asymptotic analysis

Abstract

The spectral theory of semigroup generators is a crucial tool for analysing the asymptotic properties of operator semigroups. Typically, Tauberian theorems, such as the ABLV theorem, demand extensive information about the spectrum to derive convergence results. However, the scenario is significantly simplified for positive semigroups on Banach lattices. This observation extends to the broader class of eventually positive semigroups -- a phenomenon observed in various concrete differential equations. In this paper, we investigate the spectral and asymptotic properties of eventually positive semigroups, focusing particularly on the persistently irreducible case. Our findings expand upon the existing theory of eventual positivity, offering new insights into the cyclicity of the peripheral spectrum and asymptotic trends. Notably, several arguments for positive operators and semigroups do not apply in our context, necessitating the use of ultrapower arguments to circumvent these challenges.

arXiv comment27 pages. This is version 2, accepted to appear in Semigroup Forum. Compared to version 1, statement of Theorem 5.4 has been simplified, proof of Proposition 6.1 has been slightly modified for clarity, statement of Lemma 6.2 is modified, and Example 6.9 has been added. Furthermore, the bibliography has been updated

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