A note on the fixed space of positive contractions

TL;DR

This paper investigates the structure of fixed spaces of positive contractions in Banach lattices, establishing conditions for their lattice subspace nature, and explores implications for spectral cyclicity, while presenting counterexamples and open problems.

Contribution

It introduces new conditions under which fixed spaces of positive contractions form lattice subspaces in Banach lattices, extending spectral cyclicity results and highlighting limitations with counterexamples.

Findings

Fixed spaces are lattice subspaces under certain conditions

New cyclicity results for peripheral spectra of positive operators

Counterexamples illustrate limits of the main results

Abstract

We prove that, in a large class of Banach lattices, the fixed space of each commuting family of positive linear contractions is a lattice subspace. As consequences, new cyclicity results for the peripheral point spectra of positive operators and semigroups are derived; we also pose an open problem that naturally occurs in this context. Finally, a variety of counterexamples is presented to point out some limits of our results.