Hostname: page-component-65b85459fc-s9f5t Total loading time: 0 Render date: 2025-10-20T02:24:32.122Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterization of numerical ranges for antilinear operators

Part of: General theory of linear operators Special classes 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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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).

References

Amara, Z. and Oudghiri, M., $C$ -normality of rank-one perturbations of normal operators . Linear Multilinear Algebra 71(2023), no. 15, 2426–2440.Google Scholar
Bala, N., Dhara, K., Sarkar, J., and Sensarma, A., Idempotent, model, and Toeplitz operators attaining their norms . Linear Algebra Appl. 622(2021), 150–165.Google Scholar
Benhida, C., Chō, M., Ko, E., and Lee, J. E., Characterizations of a complex symmetric truncated backward shift type operator matrix and its transforms . Results Math. 78(2023), no. 1, Article no. 13. 18 pp.Google Scholar
Benhida, C., Kliś-Garlicka, K., and Ptak, M., Skew-symmetric operators and reflexivity . Math. Slovaca 68(2018), no. 2, 415–420.Google Scholar
Bercovici, H. and Timotin, D., Truncated Toeplitz operators and complex symmetries . Proc. Amer. Math. Soc. 146(2018), no. 1, 261–266.Google Scholar
Bu, Q. and Zhu, S., The Weyl-von Neumann theorem for skew-symmetric operators . Ann. Funct. Anal. 14(2023), no. 2, Article no. 43, 12 pp.Google Scholar
Chō, M., Hur, I., and Lee, J. E., Numerical ranges of conjugations and antilinear operators on Banach spaces . Filomat 35(2021), no. 8, 2715–2720.Google Scholar
Conway, J. B., A course in operator theory, Graduate Studies in Mathematics, 21, American Mathematical Society, Providence, RI, 2000.Google Scholar
Garica, S. R., Approximate antilinear eigenvalues problems and related inequalities . Proc. Amer. Math. Soc. 136(2008), no. 1, 171–179.Google Scholar
Garcia, S. R. and Putinar, M., Complex symmetric operators and applications . Trans. Amer. Math. Soc. 358(2006), no. 3, 1285–1315.Google Scholar
Garcia, S. R. and Wogen, W. R., Complex symmetric partial isometries . J. Funct. Anal. 257(2009), no. 4, 1251–1260.Google Scholar
Guo, K., Ji, Y., and Zhu, S., A ${C}^{\ast }$ -algebra approach to complex symmetric operators . Trans. Amer. Math. Soc. 367(2015), no. 10, 6903–6942.Google Scholar
Hu, L., Li, S., and Yang, R., $C$ -normal composition operators on ${H}^2$ $.$ Ann. Math. Sci. Appl. 9(2024), no. 2, 505–525.Google Scholar
Hu, X. H., Dong, X. T., and Zhou, Z. H., Complex symmetric monomial Toeplitz operators on the unit ball . J. Math. Anal. Appl. 492(2020), no. 2, Article no. 124490.16 pp.Google Scholar
Hur, I. and Lee, J. E., Numerical ranges of conjugations and antilinear operators . Linear Multilinear Algebra 69(2021), no. 16, 2990–2997.Google Scholar
Ko, E., Lee, J. E., and Lee, M. J., On properties of $C$ -normal operators . Banach J. Math. Anal. 15(2021), no. 4, Article no. 65.17 pp.Google Scholar
Kołaczek, D. and Müller, V., Numerical ranges of antilinear operators . Integral Equ. Oper Theory 96(2024), no. 2, Article no. 17. 15 pp.Google Scholar
Li, C. G. and Zhu, S., Skew symmetric normal operators . Proc. Amer. Math. Soc. 141(2013), no. 8, 2755–2762.Google Scholar
Liu, T., Xie, X., and Zhu, S., An interpolation problem for conjugations II . Mediterr. J. Math. 19 (2022), no. 4, Article no. 153. 13 pp.Google Scholar
Liu, T., Zhao, J., and Zhu, S., Reducible and irreducible approximation of complex symmetric operators . J. Lond. Math. Soc. 100(2019), no. 1, 341–360.Google Scholar
Mashreghi, J., Ptak, M., and Ross, W. T., Conjugations of unitary operators, I. Anal. Math. Phys. 14(2024), no. 3, Article no. 62. 31 pp.Google Scholar
Murphy, G. J., ${C}^{\ast}$ -algebras and operator theory. Academic Press, Inc., Boston, MA, 1990.Google Scholar
Ptak, M., Simik, K., and Wicher, A., $C$ -normal operators . Electron. J. Linear Algebra 36(2020), 67–79.Google Scholar
Waleed, N. S., On an example of a complex symmetric composition operator on ${H}^2(D)$ . J. Funct. Anal. 269(2015), no. 6, 1899–1901.Google Scholar
Wang, C., Zhao, J., and Zhu, S., Remarks on the structure of $C$ -normal operators . Linear Multilinear Algebra 70(2022), no. 9, 1682–1696.Google Scholar
Wang, M., Wu, Q., and Han, K., Complex symmetry of Toeplitz operators over the bidisk . Acta Math. Sci. Ser. B. 43(2023), no. 4, 1537–1546.Google Scholar
Zagorodnyuk, S. M., On a $J$ -polar decomposition of a bounded operator and matrices of $J$ -symmetric and $J$ -skew-symmetric operators . Banach J. Math. Anal. 4(2010), no. 2, 11–36.Google Scholar
Zagorodnyuk, S. M., On the complex symmetric and skew-symmetric operators with a simple spectrum . SIGMA Symmetry Integr. Geom. Methods Appl. 7(2011), Paper 016. 9 pp.Google Scholar
Zhu, S., Approximation of complex symmetric operators . Math. Ann. 364(2016), nos. 1–2, 373–399.Google Scholar
Zhu, S. and Li, C. G., Complex symmetry of a dense class of operators . Integral Equ. Oper. Theory 73(2012), no. 2, 255–272.Google Scholar
Zhu, S. and Li, C. G., Complex symmetric weighted shifts . Trans. Amer. Math. Soc. 365(2013), no. 1, 511–530.Google Scholar