Cancellation properties of composition operators on Bergman spaces

Abstract

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 phi(i) are distinct and none of C-phi i is compact, then (C-phi 1 - C-phi 2) - (C-phi 3 - C-phi 1) is compact if and only if both (C-phi 1 - C-phi 2) and (C-phi 3 - C-phi 1) are compact. (C) 2015 Elsevier Inc. All rights reserved.