The canonical solution operator to partial derivative restricted to Bergman spaces

We first show that the canonical solution operator to <(<partial derivative >)over bar> 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 C the canonical solution operator to <(<partial derivative>)over bar> restricted to (0; 1)-forms wi th 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 (Cn), n>1; the corre sponding operator fails to be a Hilbert-Schmidt operator. We also indicate a connection with the theory of Hankel operators.