Operator symmetric moduli and sharp triangle inequalities

TL;DR

This paper compares different operator moduli, establishes sharp inequalities for unitarily invariant norms, and provides explicit examples to confirm optimal constants, addressing open questions in the field.

Contribution

It introduces and compares symmetrized operator moduli, derives sharp triangle inequalities, and confirms the optimality of constants, solving open problems by Bourin and Lee.

Findings

Sharp equivalence constants among operator moduli for unitarily invariant norms.

Optimal triangle inequalities with Schatten p-norm bounds.

Explicit low-dimensional examples confirming the best possible constants.

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}.