Daugavet and diameter two properties in Orlicz-Lorentz spaces

TL;DR

This paper characterizes diameter two properties and the Daugavet property in Orlicz-Lorentz spaces, linking these geometric features to conditions on Orlicz functions and their duals, with implications for the structure of these spaces.

Contribution

It provides a comprehensive characterization of diameter two properties and the Daugavet property in Orlicz-Lorentz spaces based on Orlicz function conditions, extending understanding of their geometric structure.

Findings

Spaces with non-$ riangle_2$-condition have strong diameter two properties.

Orlicz-Lorentz spaces with the Daugavet property are isometric to $L_1$ under certain conditions.

K"othe duals of these spaces lack local diameter two property when equipped with N-functions at infinity.

Abstract

In this article, we study the diameter two properties (D2Ps), the diametral diameter two properties (diametral D2Ps), and the Daugavet property in Orlicz-Lorentz spaces equipped with the Luxemburg norm. First, we characterize the Radon-Nikod\'ym property of Orlicz-Lorentz spaces in full generality by considering all finite real-valued Orlicz functions. To show this, the fundamental functions of their K\"othe dual spaces defined by extended real-valued Orlicz functions are computed. We also show that if an Orlicz function does not satisfy the appropriate $\Delta_2$-condition, the Orlicz-Lorentz space and its order-continuous subspace have the strong diameter two property. Consequently, given that an Orlicz function is an N-function at infinity, the same condition characterizes the diameter two properties of Orlicz-Lorentz spaces as well as the octahedralities of their K\"othe dual spaces. The Orlicz-Lorentz function spaces with the Daugavet property and the diametral D2Ps are isometrically isomorphic to $L_1$ when the weight function is regular. In the process, we observe that every locally uniformly nonsquare point is not a $\Delta$-point. This fact provides another class of real Banach spaces without $\Delta$-points. As another application, it is shown that for Orlicz-Lorentz spaces equipped with the Luxemburg norm defined by an N-function at infinity, their K\"othe dual spaces do not have the local diameter two property, and so as other (diametral) diameter two properties and the Daugavet property.