Cancellation properties of composition operators on Bergman spaces

The compact difference of two composition operators on the Bergman spaces over the unit disc is characterized in [11] in terms of certain cancellation property of the inducing maps at every “bad” boundary points, which make each single composition operator not to be compact. In this paper, we completely characterize the compactness of a linear combination of three composition operators on the Bergman space. As one consequence of this characterization, we show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that if φiφi are distinct and none of CφiCφi is compact, then (Cφ1−Cφ2)−(Cφ3−Cφ1)(Cφ1−Cφ2)−(Cφ3−Cφ1) is compact if and only if both (Cφ1−Cφ2)(Cφ1−Cφ2) and (Cφ3−Cφ1)(Cφ3−Cφ1) are compact.