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A characterization of numerical ranges for antilinear operators

Part of: Special classes of linear operators General theory of linear operators

Published online by Cambridge University Press:  14 April 2025

Boting Jia Open the ORCID record for Boting Jia [Opens in a new window]  and
Ting Liu
Boting Jia
Affiliation:
School of Statistics, Jilin University of Finance and Economics, Changchun 130117, P. R. China e-mail: botingjia@163.com
Ting Liu*
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P. R. China
*
e-mail: liut037@nenu.edu.cn
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Abstract

This article aims to study the problem of determining the numerical ranges of antilinear operators on complex Hilbert spaces. First, we provide a concrete description of the numerical range $W(R)$ for every bounded antilinear operator R on a complex Hilbert space $\mathcal {H}$, solving the preceding problem. Second, given a bounded linear operator T on $\mathcal {H}$, we determine the possible value of the numerical radius $w(CT)$ of $CT$ when C ranges over the collection of all conjugations on $\mathcal {H}$.

Keywords

Numerical rangesconjugationscomplex symmetric operatorsskew symmetric operatorsToeplitz operators

MSC classification

Primary: 47A12: Numerical range, numerical radius
Secondary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The second author is the corresponding author and was partially supported by the National Natural Science Foundation of China (Grant No. 12101114). The first author was partially supported by Jilin Provincial Education Department (Grant No. JJKH20240193KJ).

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