Extensions of Perron-Frobenius Theory

TL;DR

This paper extends Perron-Frobenius theory to irreducible operators on arbitrary Banach lattices, broadening the scope of classical and recent operator inequalities.

Contribution

It generalizes existing Perron-Frobenius results from positive matrices and operators on $L_p$ spaces to operators on general Banach lattices.

Findings

Extended Perron-Frobenius results to arbitrary Banach lattices.

Provided conditions under which operator equality holds.

Broadened applicability of Perron-Frobenius theory in functional analysis.

Abstract

The classical Perron-Frobenius theory asserts that for two matrices $A$ and $B$, if $0\leq B \leq A$ 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.