Composition followed by differentiation between weighted Banach spaces of holomorphic functions

Abstract

Let \({\phi}\) be an analytic self-map of the open unit disk \({\mathbb{D}}\) in the complex plane. This map induces a composition operator followed by differentiation \({DC_{\phi}}\) acting between weighted Banach spaces of holomorphic functions. We give a characterization for such an operator to be bounded resp. compact in terms of the involved weights as well as the function \({\phi}\) .