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Weak nearly uniform smoothness of the ψ-direct sums (X1 O…O XN)ψ

Kato M., Tamura T.

Commentationes Mathematicae

2012

Vol. 52, [Z] 2

171--198

We shall characterize the weak nearly uniform smoothness of the ψ-direct sum (X1 O…O XN)ψof N Banach spaces X1,..., XN, where ψ is a convex function satisfying certain conditions on the convex set [formula]. To do this, a class of convex functions which yield l1-like norms will be introduced. We shall apply our result to the fixed point property for nonexpansive mappings (FPP). In particular, an example which indicates that there are plenty of Banach spaces with FPP failing to be uniformly non-square will be presented.

A new geometrical constant of Banach spaces and the uniform normal structure

Mitani K-I., Saito K-S.

Commentationes Mathematicae

2009

Vol. 49, [Z] 1

3-13

We introduce and study a new geometrical constant γXψ of a Banach space X, by using the notion of ψ-direct sum given in [Y. Takahashi, M. Kato and K.-S. Saito, J. Inequal. Appl. 7 (2002), 179-186]. At rst, we characterize uniform non- squareness in terms γXψ Moreover, we consider Banach spaces having uniform normal structure.

Weak nearly uniform soothness and worth property of [fi]-direct sums of Banach spaces

Kato M., Tamura T.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2006

Vol. 46, [Z] 1

113-129

We shall characterize the weak nearly uniform smoothness of the fi-direct sum X Y of Banach spaces X and Y . The Schur and WORTH properties will be also characterized. As a consequence we shall see in the [...]-sums of Banach spaces there are many examples of Banach spaces with the fixed point property which are not uniformly non-square.