The C-numerical range and unitary dilations

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 \(\textrm{tr}\,(CX\,^*\,TX)\), where X is a partial isometry satisfying \(X^*X = I_n\). It is shown that

$$\begin{aligned} \textbf{cl}(W_C(T)) = \cap \{\textbf{cl}(W_C(U)): U \hbox { is a unitary dilation of } T\} \end{aligned}$$

for any contraction T if and only if C is a rank one normal matrix.