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Triangle inequalities for the operator symmetric modulus

Jean-Christophe Bourin, Eun-Young Lee

TL;DR

The paper addresses the failure of the triangle inequality for the symmetric modulus $|Z|_{\mathrm{sym}}$ with the operator norm and develops a robust framework of unitary-orbit based bounds for sums of matrices. It proves a main inequality with a free parameter $\beta>0$ and a unitary $V$ that controls $|\sum X_k|_{\mathrm{sym}}$ via a combination of $\sum |X_k|_{\mathrm{sym}}$ and a symmetric sum of moduli, yielding a universal $\sqrt{2}$-factor bound for any symmetric norm and trace-type corollaries. The work further introduces polar Hermitian sums to sharpen bounds when the sum is polar Hermitian or an involution, and proves a sharp $1/4$-constant inequality for sums of normal matrices, with consequences for Schur products and normal-sum bounds. It also offers an alternative route to Zhang's triangle inequality through an equivalence of Thompson-type forms and discusses connections to majorisation, matrix geometric means, and potential extensions to operator algebras, highlighting both theoretical significance and avenues for refinement.

Abstract

We study the operator symmetric modulus (|Z|+|Z^*|)/2 for matrices Z. Several triangle type inequalities are given.

Triangle inequalities for the operator symmetric modulus

TL;DR

The paper addresses the failure of the triangle inequality for the symmetric modulus with the operator norm and develops a robust framework of unitary-orbit based bounds for sums of matrices. It proves a main inequality with a free parameter and a unitary that controls via a combination of and a symmetric sum of moduli, yielding a universal -factor bound for any symmetric norm and trace-type corollaries. The work further introduces polar Hermitian sums to sharpen bounds when the sum is polar Hermitian or an involution, and proves a sharp -constant inequality for sums of normal matrices, with consequences for Schur products and normal-sum bounds. It also offers an alternative route to Zhang's triangle inequality through an equivalence of Thompson-type forms and discusses connections to majorisation, matrix geometric means, and potential extensions to operator algebras, highlighting both theoretical significance and avenues for refinement.

Abstract

We study the operator symmetric modulus (|Z|+|Z^*|)/2 for matrices Z. Several triangle type inequalities are given.
Paper Structure (6 sections, 18 theorems, 56 equations)

This paper contains 6 sections, 18 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The symmetric modulus
  3. Proof of Theorem \ref{['thsym']} and polar Hermitians
  4. A sharp inequality for normal matrices
  5. Around Zhang's triangle inequality
  6. Concluding remarks

Key Result

Theorem 1.1

Let $A,B\in\mathbb{M}_n$. Then, for some unitaries $U,V\in\mathbb{M}_n$,

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • ...and 16 more