On the higher rank numerical range of the shift operator

Abstract

For any n-by-n complex matrix T and any $1⩽k⩽n$, let ${\Lambda }_k(T)$ the set of all $λ\in \C$ 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 $\cos \frac{k\pi }{n+1}$ if $1<k⩽\left[\frac{n+1}{2}\right]$ and the empty set if $\left[\frac{n+1}{2}\right]<k⩽n$, 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.

References in corpus (1)

• Constrained von Neumann inequalities