On the Bonsall cone spectral radius and the approximate point spectrum

TL;DR

This paper investigates the spectral properties of nonlinear positively homogeneous maps on max-cones, establishing inclusion relations between the Bonsall cone spectral radius and the approximate point spectrum, with implications for max-type operators.

Contribution

It generalizes spectral radius results to nonlinear max-type operators and establishes the inclusion of the spectral radius in the approximate point spectrum.

Findings

Bonsall cone spectral radius is always in the approximate point spectrum.

The approximate point spectrum contains a possibly trivial interval.

Results apply to a broad class of nonlinear max-type operators.

Abstract

We study the Bonsall cone spectral radius and the approximate point spectrum of (in general non-linear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that the Bonsall cone spectral radius of such maps is always included in its approximate point spectrum. Moreover, the approximate point spectrum always contains a (possibly trivial) interval. Our results apply to a large class of (nonlinear) max-type operators. We also generalize a known result that the spectral radius of a positive (linear) operator on a Banach lattice is contained in the approximate point spectrum. Under additional generalized compactness type assumptions our results imply Krein-Rutman type results.