N-tuple wights noncommutative Orlicz spaces and some geometrical properties

Abstract

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