On three recent questions of Bourin and Lee on quadratic symmetric modulus and Euler-operator identity

TL;DR

This work resolves three Bourin–Lee questions on symmetric moduli and Euler-type identities in noncommutative matrix analysis. It proves the exponent bound $p\le 2$ in the unitary-orbit inequality is sharp in all dimensions $n\ge 2$ via a $2\times2$ counterexample, and constructs a compact operator for which the singular-value inequality fails for every $p>0$. It then develops Euler-type operator identities and their isometry-orbit refinements, yielding Clarkson–McCarthy type inequalities for Schatten norms and establishing unitary-orbit modulus inequalities with concrete singular-value consequences. The results provide a coherent framework linking symmetric moduli, unitary/isometry orbits, and norm inequalities, with broad implications for Schatten norms, Ky Fan bounds, and related operator inequalities.

Abstract

We answer three questions posed by Bourin and Lee on symmetric moduli and related orbit inequalities in \cite{BL26}. First, we show that the exponent $2$ in their unitary-orbit estimate for the quadratic symmetric modulus is optimal in every dimension $n\ge2$ by an explicit $2\times2$ counterexample for $p>2$.Second, we construct a compact operator for which the corresponding singular-value inequality fails for every $p>2$ (indeed for all $p>0$ at a fixed pair of indices).Finally, we obtain isometry-orbit refinements of Euler's quadrilateral identity for matrices and derive several Clarkson--McCarthy type inequalities for Schatten $p$-norms and related consequences.

Theorems & Definitions (41)

• Theorem 1.1: Bourin--Lee
• Theorem 1.3
• Remark 1.4
• Theorem 1.6
• Proposition 1.8: Euler operator identity
• Theorem 1.9
• Theorem 1.10
• Theorem 1.11
• Corollary 1.12
• Corollary 1.13