Journal of Operator Theory

Volume 46, Issue 2, Fall 2001 pp. 435-447.

Constrained unitary dilations and numerical ranges

Authors: Man-Duen Choi (1), and Chi-Kwong Li (2)
Author institution: (1) Department of Mathematics, University of Toronto, Toronto, M5S 3G3 Ontario, Canada
(2) Department of Mathematics, The College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187--8795, USA

Summary: It is shown that each contraction $A$ on a Hilbert space $\HH$, with $A+A^* \le \mu I$ for some $\mu \in \IR$, has a unitary dilation $U$ on $\HH\oplus \HH$ satisfying $U+U^* \le \mu I$. This is used to settle a conjecture of Halmos in the affirmative: The closure of the numerical range of each contraction $A$ is the intersection of the closures of the numerical ranges of all unitary dilations of $A$. By means of the duality theory of completely positive linear maps, some further results concerning numerical ranges inclusions and dilations are deduced.

Keywords: Unitary dilation, numerical range, Hilbert space, completely positive linear map

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