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The C-numerical range and unitary dilations

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Abstract

For an \(n\times n\) complex matrix C, the C-numerical range of a bounded linear operator T acting on a Hilbert space of dimension at least n is the set of complex numbers \(\textrm{tr}\,(CX\,^*\,TX)\), where X is a partial isometry satisfying \(X^*X = I_n\). It is shown that

$$\begin{aligned} \textbf{cl}(W_C(T)) = \cap \{\textbf{cl}(W_C(U)): U \hbox { is a unitary dilation of } T\} \end{aligned}$$

for any contraction T if and only if C is a rank one normal matrix.

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Acknowledgements

The author would like to thank the referee for her/his careful reading of the manuscript, and helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, College of William & Mary, Williamsburg, VA , 23185, USA

    Chi-Kwong Li

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  1. Chi-Kwong Li
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Correspondence to Chi-Kwong Li.

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In memory of Mrs. Tso-Mei Au-Yeung.

This research was supported by the Simons Foundation Grant 851334.

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Li, CK. The C-numerical range and unitary dilations. Acta Sci. Math. (Szeged) 89, 437–448 (2023). https://doi.org/10.1007/s44146-023-00071-0

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  • DOI: https://doi.org/10.1007/s44146-023-00071-0

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