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

Teng Zhang

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.

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

Abstract

We characterize the matrices for which the operator square-sum identity holds for all Schatten-class operators ; this happens exactly when is an isometry.Using this characterization, we establish Clarkson--McCarthy type inequalities for several classes of operator families, including -- estimates and mixed -- estimates.We also obtain a multivariable extension of the Ball--Carlen--Lieb -uniform convexity inequality and a weaker bound toward Audenaert's norm-compression conjecture.

Paper Structure

This paper contains 12 sections, 28 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Clarkson--McCarthy inequalities
  3. Clarkson--McCarthy type inequalities for several operators
  4. New examples
  5. Euler's identity
  6. A mean-variance identity
  7. All sign combinations
  8. A star-type identity
  9. Proof of Theorems \ref{['thm:identity_unitary']} and \ref{['thm:schatten_p_norms']}
  10. Proof of Theorem \ref{['thm:schatten_p_q_norms']}
  11. Proof of Theorem \ref{['thm:extension-BCL']}
  12. Proof of Theorem \ref{['thm:Audenaert-extension']}

Key Result

Theorem 1.1

Let $A,B\in S_p$. Then where $q$ denotes the conjugate exponent of $p$, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Moreover, e1 holds with the reverse inequality for $0<p\le 2$, and e2 holds with the reverse inequality for $2\le p<\infty$.

Theorems & Definitions (47)

  • Theorem 1.1: McCarthy
  • Theorem 1.2: Ball--Carlen--Lieb
  • Theorem 1.3: Ball--Carlen--Lieb--Heinävaara
  • Theorem 1.4: Bhatia--Kittaneh
  • Theorem 1.5: Audenaert--Hirzallah--Kittaneh--Zhang
  • Theorem 1.6: Kečkić
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 37 more