On K\"othe duals of Orlicz-Lorentz spaces

In this article, we study a number of properties of the K\"othe duals Mφ,w of Orlicz-Lorentz spaces. An explicit description of the order-continuous subspace of Mφ,w is provided. Moreover, the separability of these spaces is characterized by the growth condition Δ₂. Consequently, the K\"othe dual space Mφ,w has the Radon-Nikod\'ym property if and only if the N-function at infinity φ satisfies the appropriate Δ₂-condition. The comparison between Mφ,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.