N-Tuples of weighted noncommutative Orlicz space and some geometrical properties

Abstract In this article, we present a new concept named the N-tuples weighted noncommutative Orlicz space ⊕ j = 1 n L p , λ ( Φ j ) ( ℳ ˜ , τ ) {\oplus }_{j=1}^{n}{L}_{p,\lambda }^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ) , where L ( Φ j ) ( ℳ ˜ , τ ) {L}^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ) is the noncommutative Orlicz space. Based on the maximum principle, the Riesz-Thorin interpolation theorem of ⊕ j = 1 n L p , λ ( Φ j ) ( ℳ ˜ , τ ) {\oplus }_{j=1}^{n}{L}_{p,\lambda }^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ) is given. As applications, we obtain the Clarkson inequality and some other geometrical properties which include the uniform convexity and uniform smoothness of noncommutative Orlicz spaces L ( Φ s ) ( ℳ ˜ , τ ) , 0 < s ≤ 1 {L}^{\left({\Phi }_{s})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ),0\lt s\le 1 .