Eventually and asymptotically positive semigroups on Banach lattices

TL;DR

This paper develops a comprehensive theory of eventually positive semigroups on Banach lattices, characterizing their spectral properties, and introduces the concept of asymptotic positivity to include broader classes of examples.

Contribution

It provides spectral and resolvent characterizations of eventually positive semigroups and introduces asymptotic positivity, expanding the scope of applications to new differential operators and systems.

Findings

Characterization of eventually positive semigroups via spectral properties

Application to differential operators like Laplacian and bi-Laplacian

Introduction of asymptotic positivity for broader systems

Abstract

We develop a theory of eventually positive $C_0$-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neumann operator on $L^2$ and the Laplacian with non-local boundary conditions on $L^2$ within the one unified theory. We also introduce and analyse a weaker notion of eventual positivity which we call "asymptotic positivity", where trajectories associated with positive initial data converge to the positive cone in the Banach lattice as $t \to \infty$. This allows us to discuss further examples which do not fall within the above-mentioned framework, among them a network flow with non-positive mass transition and a certain delay differential equation.