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

TL;DR

This paper characterizes the closure of the orbit-closed $C$-numerical range for operators using submajorization and the essential numerical range, generalizing previous results for the $k$-numerical range.

Contribution

It provides an explicit description of the closure of the orbit-closed $C$-numerical range in terms of submajorization and the essential numerical range, extending prior work.

Findings

Explicit description of the closure of the orbit-closed $C$-numerical range.

Generalization of previous results for the $k$-numerical range.

Connection between submajorization and the numerical range closure.

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.