Carbery's inequality in the Schatten--von Neumann classes

Abstract

Carbery posed a question of sharpened triangle inequalities for families of operators in the Schatten--von Neumann classes $S_p$, $p\ge 2$. He established a weaker form of the desired estimate for even integer values of $p$. In the commutative setting the corresponding sharp inequality (with optimal exponent $p^{\prime }=\frac{p}{p-1}$) was recently obtained for all integer $p\ge 2$. In the present work we resolve Carbery's question completely in the non-commutative setting: we prove the sharp inequality $$\Vert \sum _j{T}_j{\Vert }_{S_p}\le \Vert ({\alpha }_{ij}^{p^{\prime }}){\Vert }_{{\ell }_2\to {\ell }_2}^{1/p^{\prime }}(\sum _j\Vert T_j{\Vert }_{S_p}^p{)}^{1/p}$$ for all $p\ge 2$ and all countable sequences of operators $(T_j)\subset S_p$, where ${\alpha }_{ij}$ are almost orthogonality coefficients. The proof is based on a block-operator reduction and a complex interpolation of the polar parts of the blocks.