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An extension of the Hausdorff-Toeplitz theorem on the numerical range


Authors: Yik Hoi Au-Yeung and Nam-Kiu Tsing
Journal: Proc. Amer. Math. Soc. 89 (1983), 215-218
MSC: Primary 15A60; Secondary 15A51, 47A12
DOI: https://doi.org/10.1090/S0002-9939-1983-0712625-4
MathSciNet review: 712625
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Abstract: Let $ {\mathcal{H}_n}$ be the set of all $ n \times n$ hermitian matrices and $ {\mathcal{U}_n}$ the set of all $ n \times n$ unitary matrices. For any $ c = ({c_1}, \ldots ,{c_n}) \in {{\mathbf{R}}^n}$ and $ {A_1}$, $ {A_2}$, $ {A_3} \in {\mathcal{H}_n}$, let $ W({A_1},{A_2},{A_3})$ denote the set

$\displaystyle \{ ({\operatorname{tr}}[c]U{A_1}{U^*},{\operatorname{tr}}[c]U{A_2}{U^*},{\operatorname{tr}}[c]U{A_3}{U^*}):U \in {\mathcal{U}_n}\} ,$

where $ [c]$ is the diagonal matrix with $ {c_1}, \ldots ,{c_n}$ as diagonal entries. In this present note, the authors prove that if $ n > 2$, then $ {W_c}({A_1},{A_2},{A_3})$ is always convex. Equivalent statements of this result, in terms of definiteness and inclusion relations, are also given. These results extend the theorems of Hausdorff-Toeplitz, Finsler and Westwick on numerical ranges, respectively.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0712625-4
Keywords: Numerical range, convexity, doubly-stochastic matrix, convex hull
Article copyright: © Copyright 1983 American Mathematical Society