Adjoints of generalized composition operators with linear fractional symbol

Abstract

Given a positive integer n and φ : U→ U, an analytic self-map of the open unit disc in the complex plane, the generalized composition operator C_φ^((n))is defined by C_φ^((n))f= f^((n) )o φ for f belonging to some Hilbert space of analytic functions on U. In this paper, we investigate some properties of generalized composition operators on the weighted Hardy spaces. Then we obtain adjoints of generalized composition operators with linear fractional symbol acting on the Hardy, Bergman and Dirichlet spaces.