On the Peripheral Spectrum of Positive Operators

TL;DR

This paper investigates the structure of the peripheral spectrum of positive operators on Banach lattices, establishing cyclicity under certain conditions and providing optimal examples, thereby generalizing existing theorems.

Contribution

It introduces new conditions ensuring the peripheral spectrum of positive operators is cyclic and offers optimal examples demonstrating the sharpness of these results.

Findings

Peripheral point spectrum is cyclic under growth and regularity conditions

Eigenspaces satisfy specific dimension estimates

Generalizes theorems of Lotz and Scheffold

Abstract

This paper contributes to the analysis of the peripheral (point) spectrum of positive linear operators on Banach lattices. We show that, under appropriate growth and regularity conditions, the peripheral point spectrum of a positive operator is cyclic and that the corresponding eigenspaces fulfil a certain dimension estimate. A couple of examples demonstrates that some of our theorems are optimal. Our results on the peripheral point spectrum are then used to prove a sufficient condition for the peripheral spectrum of a positive operator to be cyclic; this generalizes theorems of Lotz and Scheffold.