Abstract Lorentz spaces and K\"othe duality

Given a fully symmetric Banach function space $E$ and a decreasing positive weight $w$ on $I = (0, a)$, $0 < a \le \infty$, 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.