Weak nearly uniform smoothness of the \(\psi\)-direct sums \((X_1 \oplus \dots\oplus X_N)_\psi\)

Mikio Kato ; Takayuki Tamura

Commentationes Mathematicae, Tome 52 (2012) no. 2, pp. 171-198.

Résumé

We shall characterize the weak nearly uniform smoothness of the \(\psi\)-direct sum \((X_1\oplus \dots\oplus X_N)_\psi\) of \(N\) Banach spaces \(X_1,\dots,X_N\), where \(\psi\) is a convex function satisfying certain conditions on the convex set \(\Delta_N = \{(s_1 ,\dots , s_{N-1})\in \mathbb{R}_+^{N-1} : \sum_{i=1}^{N-1} s_i \leq 1\). To do this a class of convex functions which yield \(\ell_1\)-like norms will be introduced. We shall apply our result to the fixed point property for nonexpansive mappings (FPP). In particular an example will be presented which indicates that there are plenty of Banach spaces with FPP failing to be uniformly non-square.

Détail

DOI : 10.14708/cm.v52i2.5335

Mots-clés : absolute norm, convex function, \(\psi\)-direct sum of Banach spaces, weak nearly uniform smoothness, Garcı́a-Falset coefficient, Schur property, fixed point property