The complex Lu-Wenzel commutator conjecture is proven for normal, rank-one, and 2x2/3x3 matrices, but the proof of the unifying equivalence contains a gap.
Bhatia, Matrix analysis , Springer, New York (1997)
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On some conjectures by Lu and Wenzel
The complex Lu-Wenzel commutator conjecture is proven for normal, rank-one, and 2x2/3x3 matrices, but the proof of the unifying equivalence contains a gap.