Spectral Radii and Collatz–Wielandt Numbers for Homogeneous Order-preserving Maps and the Monotone Companion Norm

Abstract

It is well known that an ordered normed vector space X with normal cone X₊ has an order-preserving norm that is equivalent to the original norm. Such an equivalent order-preserving norm is given by

$$ \begin{array}{lll} \# x\# = \max \left\{ {d\left( {x,\,X_ + } \right),\,d\left( {x,\, - X_ + } \right)} \right\},\,x \in X. \end{array} $$

This paper explores the properties of this norm and of the half-norm ψ(x) = d(x,–X+) independently of whether or not the cone is normal. We use ψ to derive comparison principles for the solutions of abstract integral equations and compare Collatz–Wielandt numbers, bounds, and order-spectral radii for order-preserving homogeneous maps and give conditions for a local upper Collatz–Wielandt radius to have a lower positive eigenvector. For illustration, we consider a rank-structured population with mating.