  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.
