On the higher rank numerical range of the shift operator

TL;DR

This paper characterizes the higher rank numerical range of the n-dimensional shift operator, revealing it as a centered disc with radius depending on k, extending previous classical results and exploring nilpotent operators.

Contribution

It provides a complete description of the higher rank numerical range for the shift operator, extending classical results and analyzing nilpotent operators.

Findings

Higher rank numerical range of shift operator is a centered disc with radius cos(kπ/(n+1)).

The higher rank numerical range is empty for certain k values.

Results extend and refine previous classical numerical range findings.

Abstract

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.