On commuting and semi-commuting positive operators

Abstract

Let K K be a positive compact operator on a Banach lattice. We prove that if either [ K ⟩ [K\rangle or ⟨ K ] \langle K] is ideal irreducible, then [ K ⟩ = ⟨ K ] = L + ( X ) ∩ { K } ′ [K\rangle =\langle K]=L_+(X)\cap \{K\}’ . We also establish the Perron-Frobenius Theorem for such operators K K . Finally, we apply our results to answer questions posed by Abramovich and Aliprantis (2002) and Bračič et al. (2010).