Abstract
In this paper we study the square roots of unitary weighted composition operators on the Hardy space \(H^2\), i.e., \(W_{f,\varphi }\) when \(W_{g,\psi }=W_{f,\varphi }^{2}\) is unitary. In particular, we provide the explicit forms of the symbol functions \(\varphi \) and f of such \(W_{f,\varphi }\). Moreover, we show that the square roots \(W_{f,\varphi }\) of a unitary weighted composition operator are normal. Finally, we investigate several properties of such \(W_{f,\varphi }.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statement
Data sharing is not applicable to this article as no data were generated or analysed in this study.
References
Bourdon, P., Narayan, S.K.: Normal weighted composition operators on the Hardy space \(H^{2}({\mathbb{D}})\). J. Math. Anal. Appl. 367, 278–286 (2010)
Cowen, C.C.: Composition operators on \(H^{2}\). J. Oper. Theory 9, 77–106 (1983)
Cowen, C.C.: Linear fractional composition operators on \(H^{2}\). Integral Equ. Oper Theory 11, 151–160 (1988)
Cowen, C.C., Jung, S., Ko, E.: Normal and cohyponormal weighted composition operators on \(H^2\). Oper. Theory Adv. Appl. 240, 69–85 (2014)
Cowen, C.C., Ko, E.: Hermitian weighted composition operators on \(H^{2}\). Trans. Am. Math. Soc. 362, 5771–5801 (2010)
Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)
Deddens, J.A.: Analytic Toeplitz and composition operators. Can. J. Math. 24, 859–865 (1972)
Greene, R.E., Krantz, S.G.: Function Theory of One Complex Variable. Grad. Studies 40. Amer. Math. Soc. (2002)
Matache, V.: Isometric weighted composition opearators. N. Y. J. Math. 20, 711 (2014)
Ponnusamy, S., Silverman, H.: Complex Variables with Applications. Birkhauser, Basel (2006)
Acknowledgements
The authors wish to thank the referee for a careful reading and valuable comments for the original draft.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Turgay Kaptanoglu.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Andreas Seeger, Franz Luef and Serap Oztop.
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1A6A1A11051177) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1F1A1058633). The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2020R1I1A1A01065346).
Rights and permissions
About this article
Cite this article
Kim, Y., Ko, E. Characterizations of Square Roots of Unitary Weighted Composition Operators on \(H^2\). Complex Anal. Oper. Theory 16, 14 (2022). https://doi.org/10.1007/s11785-021-01193-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-021-01193-5