Partial matrix contractions and intersections of matrix balls (Q1200562)

For a rectangular matrix \(C\), let \(s_ 1(C) \geq s_ 2(C) \geq \dots\) denote the singular values of \(C\), let \(\| C\| = s_ 1(C)\), and when \(\| C\| \leq 1\), define \(D_ C = \sqrt{I - C^* C}\). Fix \(n \times k\) \(A\) and \(p \times m\) \(B\) and let \(\mathcal M\) be the set of all \(X\) such that there exists \(Y\) for which \(\| \left[ {A\atop Y} {X\atop B}\right]\| \leq 1\). Then \(\mathcal M\) is the intersection of the closed matrix balls \(D_{A^*}{\mathcal K}\) and \({\mathcal K}D_ B\), where \(\mathcal K\) is the set of all \(n \times m\) \(C\) such that \(\| C\| \leq 1\). The authors describe the matrix balls inside \(\mathcal M\) of maximal volume. Using this they prove that \(\mathcal M\) itself being a matrix ball is equivalent to conditions whose equivalence was established by \textit{Yu. L. Shmul'yan} [Integral Equations Oper. Theory 13, 864-882 (1990; Zbl 0745.47034)] and to another condition: \(s_ n(A) \geq s_ 1(B)\) or \(s_ m(B) \geq s_ 1(A)\). In certain cases, as when \(n\) or \(m\) equals 1, there is a unique maximal volume ball in \(\mathcal M\).