Diameter of weak neighborhoods and the Radon-Nikodym property in Orlicz-Lorentz spaces

Abstract

Given an Orlicz $N$-function $\phi$ and a positive decreasing weight $w$, we present criteria of the diameter two property and of the Radon-Nikodým property in Orlicz-Lorentz function and sequence spaces ${\Lambda }_{\phi ,w}$ and ${\lambda }_{\phi ,w}$. We show that in the spaces ${\Lambda }_{\phi ,w}$ or ${\lambda }_{\phi ,w}$ equipped with the Luxemburg norm, the diameter of any relatively weakly subset of the unit ball in these spaces is two if and only if $\phi$ does not satisfy the appropriate ${\Delta }_2$ condition, while they have the Radon-Nikodým property if and only if $\phi$ satisfies the appropriate ${\Delta }_2$ condition.

arXiv comment12 pages. Typos are fixed