The Szlenk index and the fixed point property under renorming

Abstract

Assume that X is a Banach space such that its Szlenk index Sz X is less than or equal to the first infinite ordinal ω. We prove that X can be renormed in such a way that X with the resultant norm satisfies R X < 2, where R · is the García-Falset coefficient. This leads us to prove that if X is a Banach space which can be continuously embedded in a Banach space Y with Sz Y ≤ ω, then, X can be renormed to satisfy the w-FPP. This result can be applied to Banach spaces which can be embedded in C K , where K is a scattered compact topological space such that K ω ∅. Furthermore, for a Banach space X, ·, we consider a distance in the space P of all norms in X which are equivalent to · for which P becomes a Baire space. If Sz X ≤ ω, we show that for almost all norms in the sense of porosity in P, X satisfies the w-FPP. For general reflexive spaces independently of the Szlenk index, we prove another strong generic result in the sense of Baire category.