Continuity of the cone spectral radius

Abstract:

This paper concerns the question whether the cone spectral radius $r_C(f)$ of a continuous compact order-preserving homogenous map $f\colon C\to C$ on a closed cone $C$ in Banach space $X$ depends continuously on the map. Using the fixed point index we show that if there exists $0<a_1<a_2<a_3<\ldots$ not in the cone spectrum, $\sigma _C(f)$, and $\lim _{k\to \infty } a_k = r_C(f)$, then the cone spectral radius is continuous. An example is presented showing that if such a sequence $(a_k)_k$ does not exist, continuity may fail. We also analyze the cone spectrum of continuous order-preserving homogeneous maps on finite dimensional closed cones. In particular, we prove that if $C$ is a polyhedral cone with $m$ faces, then $\sigma _C(f)$ contains at most $m-1$ elements, and this upper bound is sharp for each polyhedral cone. Moreover, for each nonpolyhedral cone there exist maps whose cone spectrum is infinite.