Composition-differentiation operators acting on certain Hilbert spaces of analytic functions

Abstract

We study composition-differentiation operators acting on the Bergman and Dirichlet space of the unit disk. We first characterize the compactness of this operator on weighted Bergman spaces. We shall then prove that for an analytic self-map $\varphi$ on the unit disk $\disk$, the induced composition-differentiation operator is bounded with dense range if and only if $\varphi$ is univalent and the polynomials are dense in the Bergman space on $\varphi(\disk)$.