Abstract
Let \({\Omega \subset{\mathbb C}^{d}}\) be an irreducible bounded symmetric domain of type (r, a, b) in its Harish–Chandra realization. We study Toeplitz operators \({T^{\nu}_{g}}\) with symbol g acting on the standard weighted Bergman space \({H_\nu^2}\) over Ω with weight ν. Under some conditions on the weights ν and ν 0 we show that there exists C(ν, ν 0) > 0, such that the Berezin transform \({\tilde{g}_{\nu_{0}}}\) of g with respect to \({H^2_{\nu_0}}\) satisfies:
for all g in a suitable class of symbols containing L ∞(Ω). As a consequence we apply a result in Engliš (Integr Equ Oper theory 33:426–455, 1999), to prove that the compactness of \({T^{\nu}_{g}}\) is independent of the weight ν, whenever \({g \in L^{\infty}(\Omega)}\) and ν > C where C is a constant depending on (r, a, b).
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Acknowledgments
I would like to take this opportunity to thank Prof. Wolfram Bauer who has been most generous with his time and ideas, calling my attention to the above problems. It is my pleasure to dedicate this work to Ibn L. Hassan Al-Moaammal and to my wife Aliye.
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H. Issa has been partially supported by an “Emmy-Noether scholarship” of DFG (Deutsche Forschungsgemeinschaft).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Issa, H. Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains. Integr. Equ. Oper. Theory 70, 569–582 (2011). https://doi.org/10.1007/s00020-011-1885-0
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DOI: https://doi.org/10.1007/s00020-011-1885-0