Differences of composition operators on weighted Bergman spaces

We obtain a new boundedness criterion for the difference of two composition operators from a weighted Bergman space A α p into a Lebesgue space L q ( μ ) , where 0 < q < p and α > - 1 . As a consequence, we provide a direct proof that such a bounded difference operator is necessarily compact. We also characterize compact differences of composition operators from A α p into A β q explicitly for 0 < p ≤ q and α , β > - 1 .