La propriété du point fixe dans les espaces de Banach avec base inconditionelle. (The fixed point properties in Banach spaces with unconditional bases)
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.
scientific article; zbMATH DE number 3987964
scientific article