Compactness of the solution operator to differential operators in several complex dimensions

We show that the canonical solution operator to the partial differential operators $\tfrac{{\partial ^k }}{{\partial ^k \bar z_i }}(1 \leqslant i \leqslant k)$ restricted to the Bergman-space of the unit poly-disc is not compact, while the canonical solution operator to the partial differential operators $\tfrac{{\partial ^k }}{{\partial \bar z_1 \cdots \partial \bar z_k }}$ restricted to the Bergman-space of the unit poly-disc is compact. This highlights a further (operator-theoretic) difference in the theory of one and several complex variables.