Another unitarily invariant norm attaining the minimum norm bound for commutators (Q603108)

\textit{A. Böttcher} and \textit{D. Wenzel} [Linear Algebra Appl. 403, 216--228 (2005; Zbl 1077.15020)] recently proved that for any unitarily invariant norm \(\|\cdot\|\) the quantity \(\sup \{\frac {\|XY-YX\|}{\|X\|\,\|Y\|}\): \(X\) and~\(Y\) are \(n\times n\) nonzero complex matrices\(\}=C\) is \(\geqslant \sqrt 2\), the equality attaining for the Frobenius norm. They have also asked whether the Frobenius norm is the only one having such property. In the present paper, the authors show that the equality in the above bound is also attained for the dual norm of the \((2,2)\)-norm.