On the higher rank numerical range of the shift operator

الأصل · EN

For any n-by-n complex matrix T and any 1 k n, let Λₖ(T) the set of all λ∈ such that PTP=λP for some rank-k orthogonal projection P be its higher rank-k numerical range. It is shown that if is the n-dimensional shift on ⁿ then its rank-k numerical range is the circular disc centred in zero and with radius π/n+1 if 1<k[n+1/2] and the empty set if [n+1/2]<k n, where [x] denote the integer part of x. This extends and rafines previous results of U. Haagerup, P. de la Harpe Haagerup on the classical numerical range of the n-dimensional shift onⁿ. An interesting result for higher rank-k numerical range of nilpotent operator is also established.