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 ${\rm tr}(CX^*TX)$, 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.