A supplement to the von Neumann trace inequality for singular values (Q2564922)

Let \(A\) and \(B\) be complex \(n\times n\) matrices with singular values \(\alpha_1 \geq\dots\geq \alpha_n\) and \(\beta_1\geq\dots\geq \beta_n\). The famous von Neumann trace inequality states that: \(\text{Re tr }UAVBW\leq \alpha_1\beta_1 +\dots+ \alpha_n\beta_n\), where \(U\), \(V\) and \(W\) are complex unitary matrices. The authors give a complete answer to the problems: Which diagonal elements can occur in the matrices \(UAVBW\) and which \(n\)-tuples can occur as eigenvalues.