Abstract
In this article, we study the complex symmetry of composition operators \(C_{\phi }f=f\circ \phi \) induced on the weighted Bergman spaces \(A^2_{\beta }(\mathbb {D}),\) by analytic self-maps of the unit disk. One of our main results shows that if \(C_\phi \) is complex symmetric then \(\phi \) must fix a point in \(\mathbb {D}\). From this, we prove that if \(\phi \) is neither constant nor an elliptic automorphism of \(\mathbb {D}\) and \(C_{\phi }\) is complex symmetric then \(C_{\phi }\) and \(C_{\phi }^*\) are cyclic operators. Moreover, by assuming \(\phi \) is an elliptic automorphism of \(\mathbb {D}\) which not a rotation and \(\beta \in \mathbb {N},\) we show that \(C_{\phi }\) is not complex symmetric whenever \(\phi \) has order greater than \(2(3+\beta ).\)
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Acknowledgements
This work is part of the doctoral thesis of the author. The author is grateful to Professor Sahibzada Waleed Noor for the many helpful discussions and suggestions. The author thank the anonymous referee for the many comments and suggestions to improve this article.
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Communicated by Daniel Aron Alpay.
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This article is part of the topical collection “Linear Operators and Linear Systems” edited by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz.
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Severiano, O.R. Complex Symmetry of Invertible Composition Operators on Weighted Bergman Spaces. Complex Anal. Oper. Theory 14, 59 (2020). https://doi.org/10.1007/s11785-020-01016-z
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DOI: https://doi.org/10.1007/s11785-020-01016-z