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Toeplitz Operators on Generalized Bergman Spaces

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Abstract

We consider the weighted Bergman spaces \({\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}\), where we set \({d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}\), with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.

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Authors and Affiliations

  1. Department of Mathematics, Prince of Songkla University, Hatyai, Songkhla, 90112, Thailand

    Kamthorn Chailuek

  2. Department of Mathematics, University of Notre Dame, 255 Hurley Building, Notre Dame, IN, 46556-4618, USA

    Brian C. Hall

Authors
  1. Kamthorn Chailuek
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  2. Brian C. Hall
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Correspondence to Brian C. Hall.

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Supported in part by a grant from Prince of Songkla University.

Supported in part by NSF Grant DMS-0555862.

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Chailuek, K., Hall, B.C. Toeplitz Operators on Generalized Bergman Spaces. Integr. Equ. Oper. Theory 66, 53–77 (2010). https://doi.org/10.1007/s00020-009-1734-6

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  • DOI: https://doi.org/10.1007/s00020-009-1734-6

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