Towards a Perron--Frobenius Theory for Eventually Positive Operators

TL;DR

This paper extends Perron--Frobenius spectral theory to infinite-dimensional operators that become positive after some power, establishing key spectral properties and distinctions among various notions of eventual positivity.

Contribution

It introduces multiple notions of eventual positivity in infinite dimensions and proves Perron--Frobenius type results without requiring compactness.

Findings

Spectral radius lies in the spectrum of eventually positive operators.

Conditions are provided for the spectral radius to be a positive eigenvalue.

Peripheral spectrum forms a cyclic set under general conditions.

Abstract

This article is a contribution to the spectral theory of so-called eventually positive operators, i.e.\ operators $T$ which may not be positive but whose powers $T^n$ become positive for large enough $n$. While the spectral theory of such operators is well understood in finite dimensions, the infinite dimensional case has received much less attention in the literature. We show that several sensible notions of "eventual positivity" can be defined in the infinite dimensional setting, and in contrast to the finite dimensional case those notions do not in general coincide. We then prove a variety of typical Perron--Frobenius type results: we show that the spectral radius of an eventually positive operator is contained in the spectrum; we give sufficient conditions for the spectral radius to be an eigenvalue admitting a positive eigenvector; and we show that the peripheral spectrum of an eventually positive operator is a cyclic set under quite general assumptions. All our results are formulated for operators on Banach lattices, and many of them do not impose any compactness assumptions on the operator.