  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).
