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On property (β) of Rolewicz in Köthe-Bochner sequence spaces

Henryk Hudzik, Paweł Kolwicz (2004)

Studia Mathematica

We study property (β) in Köthe-Bochner sequence spaces E(X), where E is any Köthe sequence space and X is an arbitrary Banach space. The question of whether or not this geometric property lifts from X and E to E(X) is examined. We prove that if dim X = ∞, then E(X) has property (β) if and only if X has property (β) and E is orthogonally uniformly convex. It is also showed that if dim X < ∞, then E(X) has property (β) if and only if E has property (β). Our results essentially extend and improve...

On some geometric properties of certain Köthe sequence spaces

Yunan Cui, Henryk Hudzik, Tao Zhang (1999)

Mathematica Bohemica

It is proved that if a Kothe sequence space X is monotone complete and has the weakly convergent sequence coefficient WCS ( X ) > 1 , then X is order continuous. It is shown that a weakly sequentially complete Kothe sequence space X is compactly locally uniformly rotund if and only if the norm in X is equi-absolutely continuous. The dual of the product space ( ⨁ i = 1 ∞ X i ) Φ of a sequence of Banach spaces ( X i ) i = 1 ∞ , which is built by using an Orlicz function Φ satisfying the Δ 2 -condition, is computed isometrically (i.e. the exact...

On the Banach-Mazur distance between continuous function spaces with scattered boundaries

Jakub Rondoš (2023)

Czechoslovak Mathematical Journal

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Y. Gordon (1970), we show that the constant 2 appearing in the Amir-Cambern theorem may be replaced by 3 for some class of subspaces. We achieve this by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces differs from the...

On the Banach-Stone problem

Jyh-Shyang Jeang, Ngai-Ching Wong (2003)

Studia Mathematica

Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of ℓ ₂ ∞ then T can be written as a weighted composition...

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