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

TL;DR

This paper characterizes isometries for Schatten-class operators and establishes Clarkson--McCarthy type inequalities, including $ ext{ell}_p$--$ ext{ell}_p$ and mixed $ ext{ell}_q$--$ ext{ell}_p$ estimates, with extensions and bounds related to operator convexity.

Contribution

It provides a complete characterization of matrices that preserve Schatten operator identities and introduces new inequalities and extensions in operator theory.

Findings

Characterization of matrices that preserve Schatten operator identities as isometries.

Establishment of Clarkson--McCarthy type inequalities for various operator classes.

Extension of the Ball--Carlen--Lieb convexity inequality and bounds toward Audenaert's conjecture.

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.