On property $(\beta )$ of Rolewicz in Köthe–Bochner sequence spaces

Henryk Hudzik¹ ; Paweł Kolwicz²

¹Faculty of Mathematics and Computer Science A. Mickiewicz University Umultowska 87 61-614 Poznań, Poland
²Institute of Mathematics Poznań University of Technology Piotrowo 3a 60-965 Poznań, Poland

Studia Mathematica, Tome 162 (2004) no. 3, pp. 195-212

Résumé

We study property $( \beta ) $ 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 $\mathop {\rm dim}\nolimits X=\infty $, then $E(X)$ has property $(\beta )$ if and only if $X$ has property $(\beta )$ and $E$ is orthogonally uniformly convex. It is also showed that if $\mathop {\rm dim}\nolimits X\infty $, then $E(X)$ has property $(\beta )$ if and only if $E$ has property $(\beta )$. Our results essentially extend and improve those from [14] and [15].

Détail

DOI : 10.4064/sm162-3-1

Keywords: study property beta bochner sequence spaces where sequence space arbitrary banach space question whether geometric property lifts examined prove mathop dim nolimits infty has property beta only has property beta orthogonally uniformly convex showed mathop dim nolimits infty has property beta only has property beta results essentially extend improve those