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 co on the open unit disk D , the induced composition -differentiation operator is bounded with dense range if and only if co is univalent and the polynomials are dense in the Bergman space on ohm := co ( D) .