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

We study composition-differentiation operators acting on the Bergman and Dirichlet space of the open unit disk. We first characterize the compactness of composition-differentiation operator on weighted Bergman spaces. We shall then prove that for an analytic self-map $\\varphi$ on the open unit disk $\\mathbb{D}$, 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 $\\Omega:=\\varphi(\\mathbb{D})$.