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arXiv 1004.3751math.FA
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关于移位算子的高阶数值域

On the higher rank numerical range of the shift operator

Haykel Gaaya

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摘要：

对于任何n × n复矩阵T和任何$1\leqslant k\leqslant n$，令$Λ_{k}(T)$是所有$λ\in \C$的集合，使得某个秩k正交投影$P$的$PTP=λP$是其较高的秩k数值域。证明了若$\bbS$是${\C}^{n}$上的n维移位，则它的秩k数值域是以零为中心的圆盘，若$1<k\leqslant\left[\frac{n+1}{2} \right]$则半径为$\cos\dfrac{kπ}{n+1}$，若$\left[\frac{n+1}{2} \right]<k\leqslant n$则为空集，其中$\left[x \right] $表示$x$的整数部分.这推广和改进了U. Haagerup，P. de la Harpe \cite{Haagerup}关于${\C}^{n}$上n维位移的经典数值域的研究。对于幂零算子的高阶$k$数值域也建立了一个有趣的结果.

英文摘要：

For any n-by-n complex matrix T and any $1\leqslant k\leqslant n$, let $Λ_{k}(T)$ the set of all $λ\in \C$ such that $PTP=λP$ 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 $\cos\dfrac{kπ}{n+1}$ 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.

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