For any n-by-n complex matrix T and any $1\leqslant k\leqslant n$, let $\Lambda_{ k}(T)$ the set of all $\lambda\in \C$ such that $PTP=\lambda P$ for some rank-k orthogonal projection $P$ be its higher rank-k numerical range. It is shown that if $\bbS$ is the n-dimensional shift on ${ \C}^{ n}$ then its rank-k numerical range is the circular disc centred in zero and with radius $\cos\dfrac{ k\pi}{ n+1}$ if $1<k\leqslant\left[\frac{ n+1}{ 2} \right]$ and the empty set if $\left[\frac{ n+1}{ 2} \right]<k\leqslant n$, where $\left[x \right] $ denote the integer part of $x$. This extends and rafines previous results of U. Haagerup, P. de la Harpe \cite{ Haagerup} on the classical numerical range of the n-dimensional shift on${ \C}^{ n}$. An interesting result for higher rank-$k$ numerical range of nilpotent operator is also established.
Categories: math.FA