Generalized Composition Operators on Weighted Hilbert Spaces of Analytic Functions

Let φ be an analytic self-map of the open unit disk D and g be an analytic function on D. The generalized composition operator induced by the maps g and φ is defined by the integral operator I ( g , φ ) f ( z ) =∫ 0 z f ′( φ ( ς )) g ( ς ) dς . Given an admissible weight ω , the weighted Hilbert space H ω consists of all analytic functions f such that ∥ f ∥ 2 H ω = | f (0)| 2 +∫ D | f ′( z )| 2 ω ( z ) dA ( z ) is finite. In this paper, we characterize the boundedness and compactness of the generalized composition operators on the space H ω using the ω -Carleson measures. Moreover, we give a lower bound for the essential norm of these operators.