Compactification and decompactification by weights on Bergman spaces

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 $alpha$.