# 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 “most economical” 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.

**Keywords:**Unitary dilation, numerical range, spatial matricial range, completely positive map.

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