Schatten Class Difference of Composition Operators on the Bergman Space

Abstract

We investigate the Schatten class membership of the difference of composition operators on the standard weighted Hilbert-Bergman space over the unit disk. Our main result is to characterize Schatten p-class differences of composition operators for the range p >= 2 and to provide a sufficient condition for the range p < 2. Our characterization for p >= 2 extends the known characterization for Hilbert-Schmidt differences. Our approach employs a discretization technique involving lattices and localized averaging over pseudohyperbolic disks. However, such an approach do not seem to work well for the range p < 2 and whether the sufficient condition for p < 2 is also necessary remains open.