Abstract
In this paper, we first completely characterize the complex symmetric Toeplitz operators \(T_\varphi \) on the Hardy spaces \(H^2({\mathbb {D}})\) with conjugations \({\mathcal {C}}_p^{i,j}\) and \({\mathcal {C}}_n\). Next, we give a method to determine the coefficients of \(\varphi (z)\) when \(T_\varphi \) is complex symmetric on \(H^2({\mathbb {D}})\) with the conjugation \({\mathcal {C}}_\sigma \), which partially solves a problem raised by [2]. Finally, we consider the complex symmetric Toeplitz operators \(T_\varphi \) on the weighted Bergman spaces \(A^2({\mathbb {B}}_{n})\) and the pluriharmonic Bergman spaces \(b^2({\mathbb {B}}_{n})\) with conjugations \({\mathcal {C}}_V\), where V is a symmetric permutation matrix.
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Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions, which considerably improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China [grant number 12101201].
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Communicated by Ilya Spitkovsky.
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Hu, X., Wang, C. & Xu, Z. Complex symmetric Toeplitz operators on the Hardy spaces and Bergman spaces. Ann. Funct. Anal. 15, 50 (2024). https://doi.org/10.1007/s43034-024-00352-x
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DOI: https://doi.org/10.1007/s43034-024-00352-x