A fixed-point characterization of weak compactness in Banach spaces with unconditional Schauder basis

Abstract

Let X be a Banach space with an 1-unconditional Schauder basis and without isomorphic copies of ℓ1⊕c0. We obtain an equivalent condition to weak compactness by means of a fixed-point theorem. Namely: a closed convex bounded subset C of X is weakly compact if and only if every cascading nonexpansive mapping T:C→C has a fixed point. We particularize our results when C is the closed unit ball of the Banach space X, obtaining a new characterization of reflexivity. Note that weak compactness is independent of the underlying equivalent norm and that every Banach space with an unconditional Schauder basis can be renormed to be 1-unconditional.