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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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