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.

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