N-tuple wights noncommutative Orlicz spaces and some geometrical properties

TL;DR

This paper explores the structure and geometric properties of noncommutative Orlicz spaces, establishing interpolation theorems and inequalities that advance understanding of their convexity and smoothness.

Contribution

It introduces a Riesz-Thorin interpolation theorem for N-tuple noncommutative Orlicz spaces and analyzes their geometric properties like uniform convexity and smoothness.

Findings

Established Riesz-Thorin interpolation theorem for noncommutative Orlicz spaces.

Proved Clarkson inequality in the context of these spaces.

Demonstrated uniform convexity and smoothness properties.

Abstract

This paper studies the $N$-tuple noncommutative Orlicz spaces $\bigoplus\limits_{j=1}^{n}L_{p,\lambda}^{(\Phi_{j})}(\widetilde{\mathcal{M}},\tau)$, where $L^{(\Phi_{j})}(\widetilde{\mathcal{M}},\tau)$ is noncommutative Orlicz spaces and $\widetilde{\mathcal{M}}$ is the $\tau$-measurable operators. Based on the maximum principle, we give the Riesz-Thorin interpolation theorem on $\bigoplus\limits_{j=1}^{n}L_{p,\lambda}^{(\Phi_{j})}(\widetilde{\mathcal{M}},\tau)$ . As applications, the Clarkson inequality and some geometrical properties such as uniform convexity and unform smooth of noncommutative Orlicz space.