Subdifferentials, faces, and dual matrices (Q2365719)

Let \(A=(a_{ij})\) be a complex \(m\times n\) matrix. A norm \(\|\cdot\|\) is uniformly invariant if \(\| A\|=\| PAQ\|\) for all unitary matrices \(P\) and \(Q\). Define the inner product \(\langle X,Y\rangle=\text{trace} XY^*=\sum_{i,j}x_{ij}\overline y_{ij}\) on the complex \(m\times n\) matrices. The dual norm \(\|\cdot\|^*\) to the norm \(\|\cdot\|\), \(\| A\|^*=\max_{\| X\|\leq 1}|\langle A,X\rangle|\), is equal to \(\| A\|^*=\max_{\| X\|\leq 1}\text{Re}\langle A,X\rangle\). A matrix \(G\) for which \(\| A\|^*=\max_{\| X\|\leq 1}\text{Re}\langle A,G\rangle\) is called a \(\|\cdot\|\)-dual matrix to \(A\). A characterization of the set of \(\|\cdot\|\)-dual matrices is given for arbitrary unitarily invariant norms.