Abstract Lorentz spaces and K\"othe duality

Given a fully symmetric Banach function space $E$ and a decreasing positive weight $w$ on $I=(0,a)$, $0<a\le \mathrm{\infty }$, the generalized Lorentz space ${\mathrm{\Lambda }}_{E,w}$ 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 ${\mathrm{\Lambda }}_{E,w}$ 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 $w\mathrm{.}{E}_w$. 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 ${\mathrm{\Lambda }}_{E^{\mathrm{\prime }},w}$, while the K"othe dual of ${\mathrm{\Lambda }}_{E,w}$ is $Q_{E^{\mathrm{\prime }},w}$. 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.