Lower spectral radius and spectral mapping theorem for suprema preserving mappings

TL;DR

This paper investigates the spectral properties of certain max-cone mappings in normed vector lattices, establishing a minimum principle for the lower spectral radius and proving a spectral mapping theorem for the approximate point spectrum, leading to new inequalities.

Contribution

It introduces a minimum principle for the lower spectral radius and proves a spectral mapping theorem for a class of max-cone preserving mappings, extending spectral theory in this context.

Findings

Lower spectral radius is a minimum of the approximate point spectrum.

Spectral mapping theorem holds for the approximate point spectrum.

New inequalities for Bonsall cone spectral radius of max-type kernel operators.

Abstract

We study Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We prove that the lower spectral radius of such a mapping is always a minimum value of its approximate point spectrum. We apply this result to show that the spectral mapping theorem holds for the approximate point spectrum of such a mapping. By applying this spectral mapping theorem we obtain new inequalites for the Bonsall cone spectral radius of max type kernel operators.