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Operator symmetric moduli and sharp triangle inequalities

Teng Zhang

Abstract

We compare the usual operator modulus with two symmetrized variants, the arithmetic symmetric modulus and the quadratic symmetric modulus. For every unitarily invariant norm, we determine sharp equivalence constants among these three moduli. We also establish sharp triangle-type inequalities for unitarily invariant norms, controlling sums of matrices by sums of symmetrized moduli, including optimal Schatten $p$-norm bounds and a phase transition phenomenon for the quadratic version. Explicit low-dimensional examples are provided to show that the constants are best possible. In particular, we answer two questions posed by Bourin and Lee in \cite{BL26b}.

Operator symmetric moduli and sharp triangle inequalities

Abstract

We compare the usual operator modulus with two symmetrized variants, the arithmetic symmetric modulus and the quadratic symmetric modulus. For every unitarily invariant norm, we determine sharp equivalence constants among these three moduli. We also establish sharp triangle-type inequalities for unitarily invariant norms, controlling sums of matrices by sums of symmetrized moduli, including optimal Schatten -norm bounds and a phase transition phenomenon for the quadratic version. Explicit low-dimensional examples are provided to show that the constants are best possible. In particular, we answer two questions posed by Bourin and Lee in \cite{BL26b}.
Paper Structure (17 sections, 33 theorems, 216 equations)

This paper contains 17 sections, 33 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Operator moduli and their equivalence
  3. Operator triangle inequality
  4. Triangle inequalities for unitarily invariant norms
  5. Triangle inequalities on Schatten $p$-norms
  6. Solutions on two questions of Bourin and Lee
  7. Two questions of Bourin and Lee
  8. Solution of Question \ref{['ques:BL-2.6']} for all $n\ge 3$
  9. Partial solution of Question \ref{['ques:BL-expansive']}
  10. Proofs of Theorems \ref{['thm:abs-vs-sym']}--\ref{['thm:sym-vs-qsym']} and Proposition \ref{['prop:no-constant']}
  11. Sharpness of the constant in \ref{['eq:Lee-uin']} and proof of Theorem \ref{['thm:Lee-optimal']}
  12. Sharpness of the constant $\sqrt{m}$ in \ref{['eq:Lee-uin']} for all $n\ge m$
  13. Proof of Theorem \ref{['thm:Lee-optimal']}
  14. Proofs of Theorems \ref{['thm:sum-vs-sym']}, \ref{['thm:sum-vs-qsym']} and \ref{['thm:qsym-Lee']}
  15. Partial results on Problem \ref{['prob:general-lee-problem']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

For every unitarily invariant norm $\|\cdot\|$ on $\mathbb{M}_n$ and every $Z\in\mathbb{M}_n$, Moreover, both constants $\tfrac{1}{2}$ and $1$ are sharp.

Theorems & Definitions (62)

  • Theorem 1.1: Usual vs arithmetic symmetric modulus
  • Theorem 1.2: Usual vs quadratic symmetric modulus
  • Theorem 1.3: Arithmetic vs quadratic symmetric modulus
  • Theorem 1.4: Thompson
  • Theorem 1.5: Zhang
  • Conjecture 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10: Bourin--Lee
  • ...and 52 more