N-tuple wights noncommutative Orlicz spaces and some geometrical 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.