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The Restriction Operator on Bergman Spaces

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Abstract

Motivated by questions related to the compactness of the \({\overline{\partial }}\)-Neumann operator, we study the restriction operator from the Bergman space of a domain in \(\mathbb {C}^n\) to the Bergman space of a non-empty open subset of the domain. We relate the restriction operator to the Toeplitz operator on the Bergman space of the domain whose symbol is the characteristic function of the subset. Using the biholomorphic invariance of the spectrum of the associated Toeplitz operator, we study the restriction operator from the Bergman space of the unit disc to the Bergman space of subdomains with large symmetry groups, such as horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp estimate of the norm of the restriction operator in case the domain and the subdomain are balls. We also study various operator theoretic properties of the restriction operator such as compactness and essential norm estimates.

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Acknowledgements

We thank Siqi Fu, Trieu Le, and László Lempert for helpful comments and discussions. We also thank the referee for many valuable suggestions and corrections which resulted in substantial improvements of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Central Michigan University, Mt. Pleasant, MI, 48859, USA

    Debraj Chakrabarti

  2. Department of Mathematics and Statistics, University of Toledo, Toledo, OH, 43606, USA

    Sönmez Şahutoğlu

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  1. Debraj Chakrabarti
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  2. Sönmez Şahutoğlu
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Correspondence to Debraj Chakrabarti.

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Debraj Chakrabarti was partially supported by Grant from the National Science Foundation (#1600371), a collaboration Grant from the Simons Foundation (# 316632), and also by an Early Career internal Grant from Central Michigan University.

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Chakrabarti, D., Şahutoğlu, S. The Restriction Operator on Bergman Spaces. J Geom Anal 30, 2157–2188 (2020). https://doi.org/10.1007/s12220-019-00178-3

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