Hilbert-schmidtness of weighted composition operators and their differences on Hardy spaces

Let u and φ be two analytic functions on the unit disk D such that φ(D) ⊂ D. A weighted composition operator uCφ induced by u and φ is defined on H2, the Hardy space of D, by uCφf := u • f φ for every f in H2. We obtain sufficient conditions for Hilbert-Schmidtness of uCφ on H2 in terms of function-theoretic properties of u and φ. Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on H2.