Corners of multidimensional numerical ranges

The $n$-dimensional numerical range of a densely defined linear operator $T$ on a complex Hilbert space \H is the set of vectors in $\C^n$ of the form $(< Te_1,e_1>,...,< Te_n,e_n>)$, where $e_1,...,e_n$ is an orthonormal system in \H, consisting of vectors from the domain of $T$. We prove that the components of every corner point of the $n$-dimensional numerical range are eigenvalues of $T$.