Cowen-Douglas function and its application on chaos

Abstract

In this paper, on \({\mathbb {D}}\) we define Cowen–Douglas function introduced by the Cowen–Douglas operator \(M_\phi ^*\) on Hardy space \({\mathcal {H}}^2({\mathbb {D}})\), moreover, we give a necessary and sufficient condition to determine when \(\phi\) is a Cowen–Douglas function, where \(\phi \in {\mathcal {H}}^\infty ({\mathbb {D}})\) and \(M_{\phi }\) is the associated multiplication operator on \({\mathcal {H}}^{2}({\mathbb {D}})\). Then, we give some applications of Cowen–Douglas function on chaos, such as its application on the inverse problem of chaos for \(\phi (T)\), where \(\phi\) is a Cowen–Douglas function and T is the backward shift operator on the Hilbert space \({\mathcal {L}}^2({\mathbb {N}})\).

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Acknowledgements

This work is supported by the National Nature Science Foundation of China (Grant No. 11801428). I would like to thank the referee for his/her careful reading of the paper and helpful comments and suggestions. Also, I would like to show my deepest gratitude to Prof. Xiaoman Chen and Yijun Yao for their helpful suggestions in the completion of the paper. Lastly, I shall extend my thanks to Puyu Cui for the helpful technique on the proof (1) of Theorem 3.

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Correspondence to Lvlin Luo.

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Communicated by Kehe Zhu.

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Luo, L. Cowen-Douglas function and its application on chaos. Ann. Funct. Anal. 11, 897–913 (2020). https://doi.org/10.1007/s43034-020-00061-1

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Keywords

  • Chaos
  • Hardy space
  • Rooter function
  • Cowen–Douglas function

Mathematics Subject Classification

  • 47A16
  • 47A65
  • 30J99
  • 37B99
  • 37D45