Compactness of operators on the Bergman space of the Thullen domain

We study compact operators on the Bergman space of the Thullen domain defined by $\{(z_1,z_2)\in \mathbb C^2: |z_1|^{2p}+|z_2|^2<1\}$ with $p>0$ and $p\neq 1$. The domain need not be smooth nor have a transitive automorphism group. We give a sufficient condition for the boundedness of various operators on the Bergman space. Under this boundedness condition, we characterize the compactness of operators on the Bergman space of the Thullen domain.