Elementary Proof of a generalization of the Perron-Frobenius theorem in an ordered Banach space

TL;DR

This paper provides an elementary proof of a generalized Perron-Frobenius theorem within ordered Banach spaces, establishing conditions for positive compact operators to have either zero spectral radius or a positive eigenvector.

Contribution

It introduces a simplified proof of a generalized Perron-Frobenius theorem applicable to ordered Banach spaces with closed generating positive cones.

Findings

Positive compact operators have zero spectral radius or a positive eigenvector.

The spectral radius corresponds to a positive eigenvalue.

The proof simplifies understanding of spectral properties in ordered Banach spaces.

Abstract

We work in an ordered Banach space with closed generating positive cone. We show that a positive compact operator has zero spectral radius or a positive eigenvector with the corresponding eigenvalue equal to the spectral radius.