Clarkson--McCarthy type inequalities, part I: $ll_p$--$ll_p$ and $ll_q$--$ll_p$ Schatten $p$-estimates

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.

💡 Research Summary

This paper investigates a unifying principle behind a broad family of operator inequalities known as Clarkson–McCarthy type inequalities, and uses it to derive new ℓₚ–ℓₚ and mixed ℓ_q–ℓₚ Schatten‑p norm estimates, a multivariate extension of the Ball–Carlen–Lieb 2‑uniform convexity inequality, and a partial result toward Audenaert’s norm‑compression conjecture.

The central observation is that an operator square‑sum identity \