Extensions of Perron-Frobenius Theory

The classical Perron-Frobenius theory asserts that for two matrices $A$ and $B$, if $0\le B\le A$ and $r(A)=r(B)$ with $A$ being irreducible, then $A=B$. This was recently extended in Bernik et al. (2012) to positive operators on $L_p(\mu )$ with either $A$ or $B$ being irreducible and power compact. In this paper, we extend the results to irreducible operators on arbitrary Banach lattices.