Compact double differences of composition operators on the Bergman spaces

As is well known on the weighted Bergman spaces over the unit disk, compactness of differences of two composition operators is characterized by certain cancellation property of the inducing maps at every bad boundary point, which makes each composition operator in the difference fail to be compact. Recently, the second and third authors obtained a result implying that double difference cancellation is not possible for linear combinations of three composition operators. In this paper, we obtain a complete characterization for compact double differences formed by four composition operators. Applying our characterization, we easily recover known results on linear combinations of two or three composition operators. As another application, we also show that double difference cancellation is possible for linear combinations of four composition operators by constructing an explicit example of a compact double difference formed by two noncompact differences. In spite of such an example, our characterization also shows that double difference cancellation may occur in the global sense only, and that genuine double difference cancellation is not possible in a certain local sense.