La propriété du point fixe dans les espaces de Banach avec base inconditionelle. (The fixed point properties in Banach spaces with unconditional bases)
type: Item
instance of: scholarly article
title: La propriété du point fixe dans les espaces de Banach avec base inconditionelle. (The fixed point properties in Banach spaces with unconditional bases)
zbMATH Open document ID: 0611.46018
DOI: 10.1007/BF01457867
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164275
review text: We give conditions which ensure that a Banach space X with an unconditional basis has the fixed point property (f.p.p.). The first theorem here is a quantitative result concerning the two relevant constants in connection with an unconditional basis \((e_ n)\) in \(X:\) \[ \lambda =Sup\{\| \sum _{k\in {\mathbb{N}}}a_ k\theta _ ke_ k\|;\quad \theta _ k=\pm 1,\quad \| \sum _{k}a_ ke_ k\| \leq 1\} \] \[ c=Sup\{\| \sum _{k\in F}a_ ke_ k\|;\quad F\subseteq {\mathbb{N}},\quad \| \sum _{k\in {\mathbb{N}}}a_ ke_ k\| \leq 1\}. \] We obtain that X has the f.p.p. provided \(c(\lambda +2)<4\). This condition does not include all cases where \(c=1\) because \(\lambda\) still could be equal to 2. However, if \(c=1\) and X has the alternate Banach-Saks property then X has the f.p.p.. Finally, as an application, one gets that X does not have a Banach-Saks property if there exists a contractive operator T on a weakly compact convex set K satisfying a rather technical condition (Theorem 3). Examples are given where these conditions are realized.
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 3987964
zbMATH Keywords: unconditional basis
zbMATH Keywords: alternate Banach Saks property
author: M. A. Khamsi
MaRDI profile type: Publication
cites work: A Fixed Point Free Nonexpansive Map
cites work: Q3041763
cites work: OnB-convex Banach spaces
cites work: Existence of fixed points of nonexpansive mappings in a space without normal structure
cites work: A Fixed Point Theorem for Mappings which do not Increase Distances
cites work: Unconditional bases and fixed points of nonexpansive mappings
cites work: Q4136031
Recommended article: Q4514574
Recommended article: Q3830093
Recommended article: The fixed point property and unbounded sets in Banach spaces
Recommended article: Q3125364
Recommended article: The fixed point property in Banach spaces with the NUS-property
Recommended article: The fixed point property for some uniformly nonoctahedral Banach spaces
Recommended article: Q3837063
Recommended article: The fixed point property in Banach spaces via the strict convexity and the Kadec-Klee property
Recommended article: The fixed point property under renorming in some classes of Banach spaces
Recommended article: The fixed-point property in Banach spaces containing a copy of \(c_0\)
description: scientific article; zbMATH DE number 3987964