Clarkson--McCarthy type inequalities, part I: ${\mathrm{\ell }}_p$--${\mathrm{\ell }}_p$ and ${\mathrm{\ell }}_q$--${\mathrm{\ell }}_p$ Schatten $p$-estimates

We characterize the matrices $U=(u_{ij})$ for which the operator square-sum identity $$\sum _{i=1}^m\mid \sum _{j=1}^n{u}_{ij}{A}_j{\mid }^2=\sum _{j=1}^n\mathrm{\mid }{A}_j{\mathrm{\mid }}^2$$ 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 ${\mathrm{\ell }}_p$–${\mathrm{\ell }}_p$ estimates and mixed ${\mathrm{\ell }}_q$–${\mathrm{\ell }}_p$ 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.