On some conjectures by Lu and Wenzel (Q1987011)

\textit{A. Böttcher} and \textit{D. Wenzel} conjectured in [Linear Algebra Appl. 403, 216--228 (2005; Zbl 1077.15020)] that \[\|XY-YX\|^2\le 2\|X\|^2\|Y\|^2\] for any real square matrices \(X, Y\). Here, \(\|\cdot\|\) is the Frobenius norm. It is nowadays known that this conjecture, dubbed complex BW inequality, is true for complex matrices. The authors provide a simple proof in this paper. Motivated by the DDVV inequality \[ \sum_{\alpha, \beta=1}^m \|\, [B_\alpha, B_\beta]\, \|^2 \le c\, \left(\sum_{\alpha=1}^m \|B_\alpha\|^2\right)^2, \] that involves commutators of finitely many, but only real symmetric matrices \(B_1, \dots, B_m\), \textit{Z. Lu} and \textit{D. Wenzel} [Oper. Theory: Adv. Appl. 259, 533--559 (2017; Zbl 1364.15015)] considered a unified generalization and came up with three conjectures and a question. One of the conjectures is called the Lu-Wenzel conjecture: for any \(B, B_2, \dots, B_m\in M(n,{\mathbb R})\) with (i) \(\mbox{Tr}(B_\alpha B_\beta^*) =0\) for any \(\alpha\not=\beta\); (ii) \(\mbox{Tr}(B_\alpha [B, B_\beta])=0\) for any \(2\le \alpha, \beta\le m\), there holds \[ \sum_{\alpha=2}^m \|\, [B, B_\alpha]\,\|^2\le \left( \max_{2\le \alpha \le m} \|B_\alpha\|^2 +\sum_{\alpha=2}^m \|B_\alpha\|^2\right)\|B\|^2. \] The authors add three more conjectures and show that over \(\mathbb R\), four of the six conjectures including the Lu-Wenzel conjecture are equivalent and are stronger than the other two ones. They also show that Lu-Wenzel conjecture is true for some special cases and hence all conjectures are true for those special cases.