Adjoints of linear fractional composition operators on S2(D)S2(D)

Given φ:D→Dφ:D→D, an analytic self-map of the unit disc in CC, the composition operator CφCφ is defined by Cφ(f)=f∘φCφ(f)=f∘φ for ff belonging to some Hilbert space of analytic functions on DD. We present a formula for the adjoint of linear-fractionally induced composition operators on S2(D)S2(D), a weighted Hardy space, which consists of analytic functions on the disc whose first derivative is in H2(D)H2(D).