On Köthe duals of Orlicz-Lorentz spaces

Abstract

In this article, we study a number of properties of the Köthe duals ${\mathcal{M}}_{\phi ,w}$ of Orlicz-Lorentz spaces. An explicit description of the order-continuous subspace of ${\mathcal{M}}_{\phi ,w}$ is provided. Moreover, the separability of these spaces is characterized by the growth condition ${\Delta }_2$. Consequently, the Köthe dual space ${\mathcal{M}}_{\phi ,w}$ has the Radon-Nikodým property if and only if the N-function at infinity $\phi$ satisfies the appropriate ${\Delta }_2$-condition. The comparison between ${\mathcal{M}}_{\phi ,w}$ spaces is characterized via standard orders between Orlicz functions. As applications of these results, we provide sufficient conditions for M-embedded order-continuous subspaces of Orlicz-Lorentz spaces equipped with the Luxemburg norm and prove the existence of a unique norm-preserving extension on Orlicz-Lorentz spaces equipped with the Orlicz norm.

27 pages