The Szlenk Index and the Fixed Point Property under Renorming

Tdomínguez Benavides

doi:10.1155/2010/268270

Abstract

Assume that is a Banach space such that its Szlenk index is less than or equal to the first infinite ordinal . We prove that can be renormed in such a way that with the resultant norm satisfies , where is the García-Falset coefficient. This leads us to prove that if is a Banach space which can be continuously embedded in a Banach space with , then, can be renormed to satisfy the w-FPP. This result can be applied to Banach spaces which can be embedded in , where is a scattered compact topological space such that . Furthermore, for a Banach space , we consider a distance in the space of all norms in which are equivalent to (for which becomes a Baire space). If , we show that for almost all norms (in the sense of porosity) in , 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.

Highlights

Assume that X, · is a Banach space
This paper focuses on the Renorming Theory in connection with the Fixed Point Theory
It is usually said that a Banach space X satisfies the weak Fixed Point Property w-FPP if for every convex weakly compact subset C of X, each nonexpansive mapping T : C → C has a fixed point

Summary

Introduction

Assume that X, · is a Banach space. The most common aim of the Renorming Theory is to find an equivalent norm which satisfies or which does not satisfy certain specific properties. Since C K satisfies the w-FPP 20 when K is a scattered compact topological space K such that K ω ∅, another natural question would be the following: assume that X is a Banach space which can be continuously embedded in C K for some K as above. In order to better understand the relevance of this result, note that in the metrizable case, if K ω ∅, C K is isomorphic to c0 and, there exists an equivalent norm · such that R C K , · 1 From this result and the main result in 16 , we can deduce that if a Banach space can be continuously embedded in C K , K as above, it can be renormed to satisfy the w-FPP. We finish with another strong generic result in the sense of Baire category for general reflexive spaces without any assumption on the Szlenk index

Szlenk Index and Fixed Points

A A γε ε

Genericity of the w-FPP and Szlenk Index