Sequences of Powers of Toeplitz Operators on the Hardy Space

  • Yong Chen
  • Kei Ji Izuchi
  • Young Joo LeeEmail author


We consider Toeplitz operators \(T_\varphi \) with symbol \(\varphi \) on the Hardy space \(H^2(\mathbb {D}^n)\) of the polydisk \(\mathbb {D}^n\). We characterize the bounded analytic functions \(\varphi \) on \(\mathbb {D}^n\) such that \(\Vert T^{*k}_{\varphi } f\Vert \rightarrow 0\) as \(k\rightarrow \infty \) and \(\sum ^\infty _{k=0}\Vert T^{*k}_{\varphi } f\Vert ^2\) \(<\infty \) for every \(f\in H^2(\mathbb {D}^n)\) respectively. We also give a characterization of the \(\varphi \) such that \(\sum ^\infty _{k=0}\Vert T^{*k}_{\varphi } f\Vert ^2=\infty \) for every nonzero \(f\in H^2(\mathbb {D}^n)\). As an application, we present a criterion for when a certain function can be a generator of the Hardy space. As a generalization of our results, we study the same problems on an infinite direct sum of Hardy spaces.


Toeplitz operator Hardy space Polydisk Generator 

Mathematics Subject Classification

Primary 47B35 Secondary 32A37 



The first author was supported by NSFC (11771401) and ZJNSFC(LY14A010013) and the third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A3B03933949). Also, the authors are grateful to the referee for many helpful comments.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsHangzhou Normal UniversityHangzhouPeople’s Republic of China
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaPeople’s Republic of China
  3. 3.Department of MathematicsNiigata UniversityNiigataJapan
  4. 4.Department of MathematicsChonnam National UniversityGwangjuKorea

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