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Integration Operators on Spaces of Dirichlet Series

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Abstract

We first study the Volterra operator V acting on spaces of Dirichlet series. We prove that V is bounded on the Hardy space \({\cal H}_0^p\) for any 0 < p ≤ ∞, and is compact on \({\cal H}_0^p\) for 1 <p ≤ ∞. Furthermore, we show that V is cyclic but not supercyclic on \({\cal H}_0^p\) for any 0 <p< ∞. Corresponding results are also given for V acting on Bergman spaces \({\cal H}_{w,0}^p\). We then study the Volterra type integration operators Tg. We prove that if Tg is bounded on the Hardy space \({{\cal H}^p}\), then it is bounded on the Bergman space \({\cal H}_w^p\).

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>Acknowledgements The authors would like to thank the referees for their helpful comments which improved the final version of presentation.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, P.R. China

    Jia Le Chen

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China

    Mao Fa Wang

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  1. Jia Le Chen
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  2. Mao Fa Wang
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Correspondence to Jia Le Chen.

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Conflict of Interest The authors declare no conflict of interest.

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This work was partially supported by the National Natural Science Foundation (Grant No. 12171373) of China; Chen was also supported by the Fundamental Research Funds for the Central Universities (Grant No. GK202207018) of China

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Chen, J.L., Wang, M.F. Integration Operators on Spaces of Dirichlet Series. Acta. Math. Sin.-English Ser. 39, 1919–1938 (2023). https://doi.org/10.1007/s10114-023-2442-x

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