WEIGHTED COMPOSITION OPERATORS BETWEEN BERGMAN-TYPE SPACES

Abstract

In this paper, we characterize the boundedness and compactness of weighted composition operatorsC'f = ˆfo' act- ing between Bergman-type spaces. LetD be the open unit disk in the complex planeC: Denote byH(D) the space of holomorphic functions on D: A weighted composition op- eratorC'(f)(z) = ˆ(z)f('(z)); for all z 2 D; where ' andare holomorphic functions deflned in D such that '(D) ‰ D: When ˆ = 1; we just have the composition operator C' deflned by C'(f) = fo' and when '(z) = z we have the multiplication operator Mdeflned by Mˆ(f) = ˆf: During the last century, composition operators have been studied extensively on spaces of analytic functions with the aim to explore the connection between the behavior of C' and function the- oretic properties of ': During the past few decades this subject has undergone explosive growth. As a consequence of the Littlewood Subor- dination principle (10) it is known that every analytic self map ' induces a bounded composition on Hardy and weighted Bergman spaces of the unit disk. However characterizing the compact composition operators acting on Hardy spaces of the disk was a di-cult problem. Commend- able work in this direction was done by Schwartz (14), Shapiro and Taylor (15), MacCluer and Shapiro (11) and Shapiro (16). Many other impor- tant properties of C' have also been studied on these spaces. We refer