  We characterize the symbols $\Phi$ for which there exists a weight w such
that the weighted composition operator M w C $\Phi$ is compact on the weighted
Bergman space B 2 $\alpha$. We also characterize the symbols for which there
exists a weight w such that M w C $\Phi$ is bounded but not compact. We also
investigate when there exists w such that M w C $\Phi$ is Hilbert-Schmidt on B
2 $\alpha$.
