GENERALIZED WEIGHTED COMPOSITION OPERATORS BETWEEN THE WEIGHTED BERGMAN SPACES

Let phi and u be analytic maps on the open unit disc D such that phi (D) subset of D and H (D) be the spaces of analytic functions on D. For a non-negative integer n, the generalized weighted composition operator D-phi,u(n) is defined by D-phi,u(n) f = u center dot (f ((n)) o phi), f is an element of H (D). In this paper, we characterize the boundedness of the generalized weighted composition operator D-phi,u(n) between the weighted Bergman spaces. We also compute the upper and lower bounds for the essential norm of this operator.