## 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.## References

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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