Difference of composition operators over the half-plane

We study the differences of composition operators acting on weighted Bergman spaces over the upper half-plane. In this setting not all composition operators are bounded and none are compact. The idea of joint pullback measure is used to give a Carleson measure characterization of when the difference of two composition operators is bounded or compact. Alternate characterizations, not using Carleson measures, are also given for certain large classes of the inducing maps for the operators. The relationship between angular derivatives and compact differences of composition operators is also explored, which, in particular, reveals a new phenomenon due to the upper half-plane not being bounded. Our results produce a variety of examples of distinct composition operators whose difference is compact, including examples when the individual operators are not bounded. The paper closes with a characterization of when the difference of composition operators is Hilbert-Schmidt.