Eventually positive semigroups: spectral and asymptotic analysis
Sahiba Arora
TL;DR
This work extends the spectral and asymptotic analysis of positive semigroups to the broader class of eventually positive semigroups, with a focus on persistently irreducible systems. By employing ultrapower techniques, it constructs positive pseudo-resolvents on suitable Banach lattices and derives cyclicity results for the peripheral spectrum, including under asymptotic domination. It establishes convergence patterns for uniformly eventually positive and persistently irreducible semigroups, including strong convergence to rank-one projections under suitable conditions. A non-emptiness criterion for the spectrum is provided via a smoothing assumption, broadening understanding of long-term behavior in non-compact or non-reflexive settings.
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.