Clarkson--McCarthy type inequalities, part I: $\ell_p$--$\ell_p$ and $\ell_q$--$\ell_p$ Schatten $p$-estimates

Abstract

We characterize the matrices $U=(u_{ij})$ for which the operator square-sum identity $$\sum_{i=1}^m\Big|\sum_{j=1}^n u_{ij}A_j\Big|^2=\sum_{j=1}^n|A_j|^2$$ holds for all Schatten-class operators $A_1,\ldots,A_n$; this happens exactly when $U$ is an isometry.Using this characterization, we establish Clarkson--McCarthy type inequalities for several classes of operator families, including $\ell_p$--$\ell_p$ estimates and mixed $\ell_q$--$\ell_p$ estimates.We also obtain a multivariable extension of the Ball--Carlen--Lieb $2$-uniform convexity inequality and a weaker bound toward Audenaert's norm-compression conjecture.