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

TL;DR

This paper introduces the orbit-closed $C$-numerical range, proving its convexity for selfadjoint $C$ via majorization, and extends properties of the $C$-numerical range from finite to infinite dimensions.

Contribution

It defines the orbit-closed $C$-numerical range, proves its convexity for selfadjoint $C$, and generalizes finite-dimensional properties to infinite-dimensional operators.

Findings

Orbit-closed $C$-numerical range has the same closure as the original.

Convexity of the orbit-closed $C$-numerical range for selfadjoint $C$ established.

Partial convexity results for $C$-numerical range under special conditions.

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 $C$ 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.