The canonical solution operator to $\overline {\partial }$ restricted to Bergman spaces

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.