Convexity of the orbit-closed C-numerical range and majorization

Abstract

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