The reproducing kernel thesis for difference of Hardy-space composition operators over the ball

Abstract

It has been known that for certain classes of linear operators on certain reproducing kernel Hilbert spaces, most basic properties (such as boundedness and/or compactness) are determined by the behaviour of the operators on the reproducing kernels. Such a phenomenon is now often referred to as the Reproducing Kernel Thesis (RKT). In this paper we prove RKT for differences of composition operators acting on the Hardy-Hilbert space over the ball. We also obtain a modified Carleson characterization for the difference related to RKT when the composition operators are bounded.