CONTINUITY OF THE CONE SPECTRAL RADIUS.

This paper concerns the question whether the cone spectral radius rC(f) of a continuous compact order-preserving homogenous map f : C → 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 < a1 < a2 < a3 < ... not in the cone spectrum, σC(f), and limk→∞ ak = rC(f), then the cone spectral radius is continuous. An example is presented showing that if such a sequence (ak)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 σC(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. [ABSTRACT FROM AUTHOR]