Adjoints of linear fractional composition operators on

Given φ : D → D , an analytic self-map of the unit disc in C , the composition operator C φ is defined by C φ ( f ) = f ∘ φ for f belonging to some Hilbert space of analytic functions on D . We present a formula for the adjoint of linear-fractionally induced composition operators on S 2 ( D ) , a weighted Hardy space, which consists of analytic functions on the disc whose first derivative is in H 2 ( D ) .