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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The canonical solution operator to $\overline {\partial }$ restricted to Bergman spaces
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by Friedrich Haslinger PDF
Proc. Amer. Math. Soc. 129 (2001), 3321-3329 Request permission

Abstract:

We first show that the canonical solution operator to $\overline {\partial }$ restricted to $(0,1)$-forms with holomorphic coefficients can be expressed by an integral operator using the Bergman kernel. This result is used to prove that in the case of the unit disc in $\mathbb C$ the canonical solution operator to $\overline {\partial }$ restricted to $(0,1)$-forms with holomorphic coefficients is a Hilbert-Schmidt operator. In the sequel we give a direct proof of the last statement using orthonormal bases and show that in the case of the polydisc and the unit ball in $\mathbb C^n,\ n>1,$ the corresponding operator fails to be a Hilbert-Schmidt operator. We also indicate a connection with the theory of Hankel operators.
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Additional Information
  • Friedrich Haslinger
  • Affiliation: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
  • Email: friedrich.haslinger@univie.ac.at
  • Received by editor(s): March 20, 2000
  • Published electronically: April 2, 2001
  • Communicated by: David S. Tartakoff
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 3321-3329
  • MSC (2000): Primary 32W05; Secondary 32A36
  • DOI: https://doi.org/10.1090/S0002-9939-01-05953-6
  • MathSciNet review: 1845009