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 $$\sum _{i=1}^m|\sum _{j=1}^n{u}_{ij}{A}_j{|}^2=\sum _{j=1}^n|A_j{|}^2$$ holds for all Schatten-class operators $A_1,\dots ,A_n$; 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.

v4: we change the title and add many related results. v3: fix some typos