A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones

Abstract

Several notions of spectral radius arise in the study of nonlinear order-preserving positively homogeneous self-maps of cones in Banach spaces. We give conditions that guarantee that all these notions lead to the same value. In particular, we give a Collatz-Wielandt type formula, which characterizes the growth rate of the orbits in terms of eigenvectors in the closed cone or super-eigenvectors in the interior of the cone. This characterization holds when the cone is normal and when a quasi-compactness condition, involving an essential spectral radius defined in terms of $k$-set-contractions, is satisfied. Some fixed point theorems for non-linear maps on cones are derived as intermediate results. We finally apply these results to show that non-linear spectral radii commute with respect to suprema and infima of families of order preserving maps satisfying selection properties.