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Cowen-Douglas function and its application on chaos

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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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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Lvlin Luo

  2. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, People’s Republic of China

    Lvlin Luo

  3. School of Science, Shanghai Institute of Technology, Shanghai, 201418, People’s Republic of China

    Lvlin Luo

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  1. Lvlin Luo
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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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  • DOI: https://doi.org/10.1007/s43034-020-00061-1

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