Non-compact positive operators

Abstract

This paper is concerned with positive operators acting in a partially ordered Banach space, and provides extensions of various theorems which involve operators of this kind which are in addition supposed to be compact. It is shown that any continuous linear positive map T has a positive eigenvector provided the spectrum of T contains a point with modulus greater than the radius of the essential spectrum of T : this result contains the well-known theorem of Krein-Rutman for compact operators. Various results connected to the Krein-Rutman theorem in a natural way are provided for non-compact positive linear operators, some involving k -set contractions and another which utilizes the notion of a projectionally compact operator. Two fixed point theorems for nonlinear positive operators are obtained by the use of topological degree theory for k -set contractions.