This is an announcement for the paper “Abstract Lorentz spaces and Köthe duality” by Anna Kamińskahttps://arxiv.org/find/math/1/au:+Kaminska_A/0/1/0/all/0/1, Yves Raynaudhttps://arxiv.org/find/math/1/au:+Raynaud_Y/0/1/0/all/0/1.

Abstract: Given a fully symmetric Banach function space $E$ and a decreasing positive weight $w$ on $I=(0, a), 0<a\leq\infty$, the generalized Lorentz space $\Lambda_{E, w}$ is defined as the symmetrization of the canonical copy $E_w$ of $E$ on the measure space associated with the weight. If $E$ is an Orlicz space then $\Lambda_{E, w}$ is an Orlicz-Lorentz space. An investigation of the K"othe duality of these classes is developed that is parallel to preceding works on Orlicz-Lorentz spaces. First a class of functions $M_{E, w}$, which does not need to be even a linear space, is similarly defined as the symmetrization of the space $w.E_w$. Let also $Q_{E, w}$ be the smallest fully symmetric Banach function space containing $M_{E, w}$. Then the K"othe dual of the class $M_{E, w}$ is identified as the Lorentz space $\Lambda_{E’, w}$, while the K"othe dual of $\Lambda_{E, w}$ is Q_{E’, w}$. The space $Q_{E, w}$ is also characterized in terms of Halperin's level functions. These results are applied to concrete Banach function spaces. In particular the K"othe duality of Orlicz-Lorentz spaces is revisited at the light of the new results.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1802.01728