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For a positive trace-class operator $C$ and a bounded operator $A$, weprovide an explicit description of the closure of the orbit-closed$C$-numerical range of $A$ in terms of those operators submajorized by $C$ andthe essential numerical range of $A$. This generalizes and subsumes recent workof Chan, Li and Poon for the $k$-numerical range, as well as some of our ownprevious work on the orbit-closed $C$-numerical range.

Closedness of the orbit-closed $C$-numerical range and submajorization