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Hilbert–Schmidt differences of composition operators on the Bergman space

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Abstract

In the setting of the weighted Bergman space over the unit disk, we characterize Hilbert–Schmidt differences of two composition operators in terms of integrability condition involving pseudohyperbolic distance between the inducing functions. We also show that a linear combination of two composition operators can be Hilbert–Schmidt, except for trivial cases, only when it is essentially a difference. We apply our results to study the topological structure of the space of all composition operators under the Hilbert–Schmidt norm topology. We first characterize components and then provide some sufficient conditions for isolation or for non-isolation.

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Authors and Affiliations

  1. Department of Mathematics, Korea University, Seoul, 136-713, Korea

    Boo Rim Choe & Hyungwoon Koo

  2. Faculty of Engineering, Ibaraki University, Hitachi, 316-8511, Japan

    Takuya Hosokawa

Authors
  1. Boo Rim Choe
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  2. Takuya Hosokawa
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  3. Hyungwoon Koo
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Correspondence to Boo Rim Choe.

Additional information

This research was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-314-C00012).

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Choe, B.R., Hosokawa, T. & Koo, H. Hilbert–Schmidt differences of composition operators on the Bergman space. Math. Z. 269, 751–775 (2011). https://doi.org/10.1007/s00209-010-0757-7

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  • DOI: https://doi.org/10.1007/s00209-010-0757-7

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