Two tuples of noncommutative Orlicz sequence spaces and some geometry properties

TL;DR

This paper introduces a new framework for noncommutative Orlicz sequence spaces, establishes an interpolation theorem for them, and explores their geometric properties through constants like the von Neumann-Jordan constant.

Contribution

It proposes the concept of 2-tuples of noncommutative Orlicz sequence spaces and proves an interpolation theorem using the three-line theorem.

Findings

Established Riesz-Thorin interpolation theorem for the new space class.

Derived bounds for nonsquare and von Neumann-Jordan constants.

Extended geometric analysis to noncommutative Orlicz spaces.

Abstract

The primary contribution of this study lies in proposing a new concept termed $2$-tuples of noncommutative Orlicz sequence spaces $\bigoplus\limits_{j=1}^{2}S_{\varphi_{j},p}$, where $S_{\varphi_{j}}$ denotes a noncommutative Orlicz sequence space. By leveraging the three-line theorem, we establish the Riesz-Thorin interpolation theorem for $\bigoplus\limits_{j=1}^{2}S_{\varphi_{j},p}$. As applications, we derive bound for the nonsquare and von Neumann-Jordan constant of noncommutative Orlicz space $S_{\varphi_{s}} (0<s\leq1)$, where $\varphi_{s}$ is an intermediate function.