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Weak nearly uniform soothness and worth property of ψ -direct sums of Banach spaces

Mikio Kato;  Takayuki Tamura — 2006

Commentationes Mathematicae

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

Weak nearly uniform smoothness of the ψ -direct sums ( X 1 ⊕ ⋯ ⊕ X N ) ψ

Mikio Kato;  Takayuki Tamura — 2012

Commentationes Mathematicae

We shall characterize the weak nearly uniform smoothness of the ψ -direct sum ( X 1 ⊕ ⋯ ⊕ X N ) ψ of N Banach spaces X 1 , ⋯ , X N , where ψ is a convex function satisfying certain conditions on the convex set Δ N = { ( s 1 , ⋯ , s N - 1 ) ∈ ℝ + N - 1 : ∑ i = 1 N - 1 s i ≤ 1 . To do this a class of convex functions which yield ℓ 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.

Uniform non- ℓ 1 n -ness of ℓ 1 -sums of Banach spaces

Mikio Kato;  Takayuki Tamura — 2007

Commentationes Mathematicae

We shall characterize the uniform non- ℓ 1 n -ness of the ℓ 1 -sum ( X 1 ⊕ ⋯ ⊕ X m ) 1 of a finite number of Banach spaces X 1 , ⋯ , X m . Also we shall obtain that ( X 1 ⊕ ⋯ ⊕ X m ) 1 is uniformly non- ℓ 1 m + 1 if and only if all X 1 , ⋯ , X m are uniformly non-square (note that ( X 1 ⊕ ⋯ ⊕ X m ) 1 is not uniformly non- ℓ 1 m ). Several related results will be presented.

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