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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Kolwicz P.

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On Property β of Rolewicz in Köthe–Bochner Function Spaces

Kolwicz P.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 1

75-85

It is proved that the Köthe–Bochner function space E(X) has property β if and only if X is uniformly convex and E has property β. In particular, property β does not lift from X to E(X) in contrast to the case of Köthe–Bochner sequence spaces.

Orthogonal uniform convexity in Köthe spaces and Orlicz spaces

Kolwicz P.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 4

395-411

We study a geometric property in Köthe spaces which is called orthogonal uniform convexity (UC┴). It was introduced in [19]. We prove that the class of Köthe spaces with property (UC┴) is a proper subset of the class of uniformly monotone and P-convex Köthe spaces. Next we consider connections between (UC┴) and property (β) of Rolewicz. We shown that the implication (UC┴) → (β) is true in any Köthe sequence space. Moreover, we find criteria for Orlicz function (sequence) spaces to be orthogonally uniformly convex. As a corollary we get that there holds (UC) → (UC┴) → (β) in any Köthe sequence space and the converse of any of these implications is not true. Furthermore, the implications (UC) → (β) → (UC┴) hold in any Köthe function space and the second one cannot be reversed.