On property $(\beta )$ of Rolewicz in Köthe–Bochner sequence spaces
Volume 162 / 2004
Studia Mathematica 162 (2004), 195-212
MSC: 46E40, 46B20, 46E30.
DOI: 10.4064/sm162-3-1
Abstract
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].
