Abstract
In this paper, we study weighted composition–differentiation operators on the Hardy space \(H^{2}({\mathbb {D}})\). We investigate which combinations of weights \(\psi \) and maps of the open unit disk \(\varphi \) give rise to complex symmetric weighted composition–differentiation operators with conjugation \({\mathcal {C}}\), where \({\mathcal {C}}\) is a \(M_{z}\)-commuting conjugation on \(H^{2}({\mathbb {D}})\). As an application, we find an equivalent condition for such an operator to be normal. In addition, we identify the Hermitian weighted composition–differentiation operators and we show that the class of all Hermitian weighted composition–differentiation operators on \(H^{2}({\mathbb {D}})\) is contained in the class of all \(C_{\xi , \theta }\)-symmetric weighted composition–differentiation operators, where \(\xi , \theta \in [0,2\pi ]\).
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Acknowledgements
The authors would like to thank the referees for the helpful and constructive comments and suggestions that greatly improve the quality of the manuscript. This work was supported by NSF of China (No. 11771340).
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Communicated by Jari Taskinen.
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Han, K., Wang, M. Weighted composition–differentiation operators on the Hardy space. Banach J. Math. Anal. 15, 44 (2021). https://doi.org/10.1007/s43037-021-00131-z
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DOI: https://doi.org/10.1007/s43037-021-00131-z