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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, 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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