  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 $C$ selfadjoint, 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 selfadjoint $C$ 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.
