On the Krein-Rutman theorem and beyond

TL;DR

This paper extends the Krein-Rutman theorem within Banach lattice frameworks, providing general results with constructive estimates on eigenvalues, eigenvectors, and stability, supported by diverse differential and integral operator examples.

Contribution

It offers new, general, and constructive results on the Krein-Rutman theorem, including eigenproblem solutions and stability analysis, applicable to various differential and integral operators.

Findings

Existence of solutions to the first eigentriplet problem.

Geometry of the principal eigenvalue problem.

Asymptotic stability with constructive convergence rates.

Abstract

In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with constructive estimates about: the existence of a solution to the first eigentriplet problem; the geometry of the principal eigenvalue problem; the asymptotic stability of the first eigenvector with possible constructive rate of convergence. This abstract theory is motivated and illustrated by several examples of differential, integro-differential and integral operators. In particular, we revisit the first eigenvalue problem and the asymptotic stability of the first eigenvector for: some parabolic equations in a bounded domain and in the whole space; some transport equations in a bounded or unbounded domain, including some growth-fragmentation models and some kinetic models; the kinetic Fokker-Planck equation in the torus and in the whole space; some mutation-selection models.