The $C$-numerical range and Unitary dilations

TL;DR

This paper investigates the $C$-numerical range of matrices and operators, establishing a characterization of its closure via unitary dilations, specifically when $C$ is a rank one normal matrix.

Contribution

It provides a necessary and sufficient condition for the closure of the $C$-numerical range to be described by unitary dilations, focusing on the case when $C$ is rank one normal.

Findings

The closure of the $C$-numerical range equals the intersection over unitary dilations for rank one normal $C$.

Characterization of when the $C$-numerical range can be approximated by unitary dilations.

Extension of classical numerical range results to the $C$-numerical range setting.

Abstract

For an $n\times n$ complex matrix $C$, the $C$-numerical range of a bounded linear operator $T$ acting on a Hilbert space of dimension at least $n$ is the set of complex numbers ${\rm tr}(CX^*TX)$, where $X$ is a partial isometry satisfying $X^*X = I_n$. It is shown that $${\bf cl}(W_C(T)) = \cap \{{\bf cl}(W_C(U)): U \hbox{ is a unitary dilation of } T\}$$ for any contraction $T$ if and only if $C$ is a rank one normal matrix.