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 $\mathrm{tr}(C{X}^{\ast }TX)$, where $X$ is a partial isometry satisfying $X^{\ast }X=I_n$. It is shown that $$\mathbf{cl}(W_C(T))=\cap \{\mathbf{cl}(W_C(U)):Uis\; a\; unitary\; dilation\; ofT\}$$ for any contraction $T$ if and only if $C$ is a rank one normal matrix.

11 pages