Weak nearly uniform smoothness of the $\psi$-direct sums ${(X_1\oplus \cdots \oplus X_N)}_{\psi }$

Mikio Kato, and Takayuki Tamura. "Weak nearly uniform smoothness of the $\psi $-direct sums $(X_1 \oplus \dots \oplus X_N)_\psi $." Commentationes Mathematicae 52.2 (2012): null. <http://eudml.org/doc/291694>.

@article{MikioKato2012,
abstract = {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 $\Delta _N = \lbrace (s_1 ,\dots , s_\{N-1\})\in \mathbb \{R\}_+^\{N-1\} : \sum _\{i=1\}^\{N-1\} s_i \le 1$. 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.},
author = {Mikio Kato, Takayuki Tamura},
journal = {Commentationes Mathematicae},
keywords = {absolute norm; convex function; $\psi $-direct sum of Banach spaces; weak nearly uniform smoothness; Garcı́a-Falset coefficient; Schur property; fixed point property},
language = {eng},
number = {2},
pages = {null},
title = {Weak nearly uniform smoothness of the $\psi $-direct sums $(X_1 \oplus \dots \oplus X_N)_\psi $},
url = {http://eudml.org/doc/291694},
volume = {52},
year = {2012},
}

TY - JOUR
AU - Mikio Kato
AU - Takayuki Tamura
TI - Weak nearly uniform smoothness of the $\psi $-direct sums $(X_1 \oplus \dots \oplus X_N)_\psi $
JO - Commentationes Mathematicae
PY - 2012
VL - 52
IS - 2
SP - null
AB - 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 $\Delta _N = \lbrace (s_1 ,\dots , s_{N-1})\in \mathbb {R}_+^{N-1} : \sum _{i=1}^{N-1} s_i \le 1$. 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.
LA - eng
KW - absolute norm; convex function; $\psi $-direct sum of Banach spaces; weak nearly uniform smoothness; Garcı́a-Falset coefficient; Schur property; fixed point property
UR - http://eudml.org/doc/291694
ER -