On Commuting and Semi-commuting Positive Operators

Abstract

Let $K$ be a positive compact operator on a Banach lattice. We prove that if either $[K>$ or $<K]$ is ideal irreducible then $[K>=<K]=L_{+}(X)\cap K^{\prime }$. 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 Bračič (2010).