On Commuting and Semi-commuting Positive Operators

TL;DR

This paper investigates positive compact operators on Banach lattices, establishing conditions under which certain operator inequalities imply ideal irreducibility and deriving a Perron-Frobenius type theorem, with applications to existing open questions.

Contribution

It proves that ideal irreducibility of either $[K>$ or $<K]$ implies equality of certain operator sets and extends the Perron-Frobenius theorem to these operators.

Findings

Equality of operator sets under ideal irreducibility conditions

Extension of Perron-Frobenius theorem to positive compact operators

Answers to open questions in operator theory literature

Abstract

Let $K$ be a positive compact operator on a Banach lattice. We prove that if either $[K>$ or $<K]$ is ideal irreducible then $[K>=<K]=L_+(X)\cap {K}'$. We also establish the Perron-Frobenius Theorem for such operators $K$. Finally we apply the results to answer questions in Abramovich and Aliprantis (2002) and Bra\v{c}i\v{c} (2010).