Journal of Operator Theory

Volume 63, Issue 1, Winter 2010 pp. 181-189.

Higher-rank numerical ranges and dilations

Authors: Hwa-Long Gau (1), Chi-Kwong Li (2), and Pei Yuan Wu (3)
Author institution: (1) Department of Mathematics, National Central University, Chung-Li 320, Taiwan
(2) Department of Mathematics, The College of William and Mary, Williamsburg, VA 23185, USA
(3) Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan

Summary: For any

$n$ -by-

$n$ complex matrix

$A$ and any

$k$ ,

$1\leqslant k\leqslant n$ , let

$\Lambda_k(A) = \{\lambda \in \IC: X^*AX = \lambda I_k$ for some

$n$ -by-

$k$

$X$ satisfying

$X^*X = I_k\}$ be its rank-

$k$ numerical range. It is shown that if

$A$ is an

$n$ -by-

$n$ contraction, then

$\Lambda_k(A) = \bigcap\{ \Lambda_k(U): U \hbox{ is an } (n+d_A) \mbox{-by-} (n+d_A) \hbox{ unitary dilation of } A\},$ where

$d_A=\rank (I_n-A^*A)$ . This extends and refines previous results of Choi and Li on constrained unitary dilations, and a result of Mirman on

$S_n$ -matrices.

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