The aim of this paper is to study the w-fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of w-closed subsets of the dual sphere is equivalent to the w-fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable w-fixed point property. The last geometrical notion was introduced by Fonf and Veselý as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators.

Weak∗ fixed point property in ℓ1 and polyhedrality in Lindenstrauss spaces

Emanuele, Casini;
2018

Abstract

The aim of this paper is to study the w-fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of w-closed subsets of the dual sphere is equivalent to the w-fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable w-fixed point property. The last geometrical notion was introduced by Fonf and Veselý as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators.
Nonexpansive mappings, w-fixed point property, Stability of the w- fixed point property, Lindenstrauss spaces, Polyhedral spaces, `1 space, Extension of compact operators.
Casini, EMANUELE GIUSEPPE; Miglierina, Enrico; Piasecki, Łukasz; Popescu, Roxana
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11383/2062949
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