# Weighted composition operators between Hardy and Bergman type spaces of the upper half-plane

#### Description

Let be a holomorphic map of the upper half-plane +, ' be a holomorphic self-map of + and H(+) be the space of all holomorphic functions on +. Define a linear operator W'; with inducing maps and ' as W'; f = (f '); f 2 H(+): The operator W'; is known as weighted composition operator and is a generalization of the composition operator C' defined by C'f = f ' and the multiplication operator M defined by M f = f : In this paper, we characterize bounded weighted composition operators acting between weighted Bergman and growth spaces of the upper half-plane. Under a mild condition on , we also characterize compact weighted composition operators between weighted Bergman and growth spaces of the upper half-plane. We also prove Carleson type theorem on weighted Bergman spaces of the upper half-plane and use it to characterize bounded weighted composition operators between weighted Bergman spaces

*This paper has been withdrawn.*

Weighted composition operators between Hardy and Bergman type spaces of the upper half-plane

Cupples I Room 207

Let be a holomorphic map of the upper half-plane +, ' be a holomorphic self-map of + and H(+) be the space of all holomorphic functions on +. Define a linear operator W'; with inducing maps and ' as W'; f = (f '); f 2 H(+): The operator W'; is known as weighted composition operator and is a generalization of the composition operator C' defined by C'f = f ' and the multiplication operator M defined by M f = f : In this paper, we characterize bounded weighted composition operators acting between weighted Bergman and growth spaces of the upper half-plane. Under a mild condition on , we also characterize compact weighted composition operators between weighted Bergman and growth spaces of the upper half-plane. We also prove Carleson type theorem on weighted Bergman spaces of the upper half-plane and use it to characterize bounded weighted composition operators between weighted Bergman spaces