Extreme points of the intersection and the union of some operator balls. (Q1865926)

Let \(L(\mathbb{C}^n)\) be the space of all linear operators on the \(n\)-dimensional complex Euclidean space \(\mathbb{C}^n\). For a given self-adjoint and positive operator \(R\), consider the operator subsets: \(K_R= \{A:A^*A \leq R^2\}\) and \(K_R^* =\{A:A^*A\leq R^2\}= \{A:A^*\in K_R\}\). These sets are the images of the unit ball in \(L(\mathbb{C}^n)\) under the transformations \(V\to VR\) and \(V\to RV\), so they are convex and compact. The author studies the extreme points of the intersection of union of \(K_R\) and \(K_R^*\). The extreme points of \(K_R\cap K_R^*\) are described when \(n=2\), and the extreme points of \(K_R\cup K_R^*\) are described for all \(n\).