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 $p$-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}.