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Multiplication and Toeplitz Operators on the Generalized Derivative Hardy Space

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Abstract

In this paper, we propose the generalized derivative Hardy space \(S^2_{\alpha ,\beta }(\mathbb {D})\) which consists of functions whose derivatives are in the Hardy and Bergman spaces. In particular, we state basic results for \(S^2_{\alpha ,\beta }(\mathbb {D})\) and focus on m-isometric multiplication operators. Moreover, we consider the complete Pick property in \(S_{\alpha ,\beta }^2(\mathbb {D})\) and several applications of having the complete Pick property, which is related to the multiplication operators and composition operators. Finally, we study the Toeplitz operators on \(S^2_{\alpha ,\beta }(\mathbb {D})\) and investigate a necessary and sufficient condition for the hyponormality of Toeplitz operator \(T_{\varphi }\) on \(S^2_{\alpha ,\beta }(\mathbb {D})\).

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Acknowledgements

The authors wish to thank the referees for their invaluable comments on the original draft.

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Authors and Affiliations

  1. Department of Mathematics, Ewha Womans University, Seoul, 120-750, Korea

    Eungil Ko

  2. Department of Mathematics and Statistics, Sejong University, Seoul, 143-747, Korea

    Ji Eun Lee

  3. Institute of Mathematical Sciences, Ewha Womans University, Seoul, 120-750, Korea

    Jongrak Lee

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  1. Eungil Ko
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  2. Ji Eun Lee
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  3. Jongrak Lee
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Correspondence to Ji Eun Lee.

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Communicated by Ilwoo Cho.

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E. Ko was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03931937). J. E. Lee was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (2019R1A2C1002653). J. Lee was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048620) and (2019R1A6A1A11051177).

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Ko, E., Lee, J.E. & Lee, J. Multiplication and Toeplitz Operators on the Generalized Derivative Hardy Space. Complex Anal. Oper. Theory 13, 4143–4164 (2019). https://doi.org/10.1007/s11785-019-00954-7

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  • DOI: https://doi.org/10.1007/s11785-019-00954-7

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