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Path Components of the Space of (Weighted) Composition Operators on Bergman Spaces

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Abstract

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces \(A_{\alpha }^p\) with \(p \in (0, \infty )\) and \( \alpha \in (-1, \infty )\). In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

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Acknowledgements

The authors thank the Editor and the Referee for useful remarks and comments that led to the improvement of the paper. The main part of this article has been done during Pham Trong Tien’s stay at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the institution for hospitality and support.

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Authors and Affiliations

  1. Southern Federal University, Rostov-on-Don, Russian Federation, 344090

    Alexander V. Abanin

  2. Southern Mathematical Institute, Vladikavkaz, Russian Federation, 362027

    Alexander V. Abanin

  3. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University (NTU), Singapore, 637371, Singapore

    Le Hai Khoi

  4. Faculty of Mathematics, Mechanics and Informatics, University of Science, Vietnam National University, Hanoi, Vietnam

    Pham Trong Tien

  5. TIMAS, Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam

    Pham Trong Tien

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  1. Alexander V. Abanin
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  2. Le Hai Khoi
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  3. Pham Trong Tien
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Correspondence to Le Hai Khoi.

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Abanin, A.V., Khoi, L.H. & Tien, P.T. Path Components of the Space of (Weighted) Composition Operators on Bergman Spaces. Integr. Equ. Oper. Theory 93, 5 (2021). https://doi.org/10.1007/s00020-020-02615-3

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  • DOI: https://doi.org/10.1007/s00020-020-02615-3

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