Journal of Operator Theory

Volume 38, Issue 1, Summer 1997 pp. 25-42.

Unitary dilations and numerical ranges

Authors: Pei Yuan Wu
Author institution: Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, REPUBLIC OF CHINA, E-mail: pywu@cc.nctu.edu.tw

Summary: We prove that any algebraic contraction T on a (separable) Hilbert space can be dilated to an operator of the form

$T_1 \otimes T_1 \otimes ...$ , where T_1 is a cyclic contraction on a finite-dimensional space with the same minimal polynomial as T and rank

$(1 - T_1^* T_1 ) \leqslant 1$ . As applications, we use this to determine the â€œmost economicalâ€ unitary dilations of finite-dimensional contractions and also the spatial matricial ranges of the unilateral shift.
Generalizing an example of Durszt, we give a necessary and sufficient condition on a normal contraction T such that its numerical range equals the intersection of the numerical ranges of unitary dilations of T.

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