Convexity of the orbit-closed <i>C</i>-numerical range and majorization

We introduce and investigate the orbit-closed $C$-numerical range, a natural modification of range an operator introduced for $C$ trace-class by Dirr vom Ende. Our is conservative theirs because these two sets have same closure even coincide when finite rank. Since Ende's results concerning depend only on its closure, our inherits properties, but we also establish more. For selfadjoint, Ende were able to prove that their convex, asked whether it convex without taking closure. convexity selfadjoint providing characterization in terms majorization, unlocking door plethora which generalize properties known dimensions or has Under rather special hypotheses operators, show thereby partial answer question posed