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

Abstract

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