Commutator inequalities associated with the polar decomposition (Q2781284)

Let \(A=UP\) be a polar decomposition of an \(n\times n\) complex matrix \(A\). Then for every unitarily invariant norm \(|||{\cdot}|||\) it is shown that \(||||UP-PU||||\leqslant |||A^*A-AA^*|||\leqslant \|UP+PU\|\cdot |||UP-PU|||\), where \(\|{\cdot}\|\) is the operator norm. This is a quantitative version of the well-known result that \(A\) is normal if and only if \(UP=PU\). As a related result, the following inequality is obtained: \(\|A\|^2-\|A^2\|\leqslant \|A^*A-AA^*\|\), which completes \textit{C. K. Fong}'s inequality \(\|A^A-AA^*\|\leqslant \|A\|^2\) [Linear Algebra Appl. 74, 151-156 (1986; Zbl 0588.15009)], both inequalities being sharp.