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arXiv · 2212.12149

Daugavet and diameter two properties in Orlicz-Lorentz spaces

Anna Kamińska·Han Ju Lee·Hyung-Joon Tag
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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-Nikodým property of Orlicz-Lorentz spaces in full generality by considering all finite real-valued Orlicz functions. To show this, the fundamental functions of their Köthe dual spaces defined by extended real-valued Orlicz functions are computed. We also show that if an Orlicz function does not satisfy the appropriate $Δ_2$-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 Köthe 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 $Δ$-point. This fact provides another class of real Banach spaces without $Δ$-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 Köthe dual spaces do not have the local diameter two property, and so as other (diametral) diameter two properties and the Daugavet property.

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BibTeXRIS

Anna Kamińska, Han Ju Lee, Hyung-Joon Tag. 2023-04-05. Daugavet and diameter two properties in Orlicz-Lorentz spaces. https://arxiv.org/abs/2212.12149

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