Weighted composition–differentiation operators in the uniformly closed algebra generated by weighted composition operators

Abstract

Let $$\varphi $$ be an analytic self map of the open unit disc $$\mathbb {D}$$ . Assume that $$\psi $$ is an analytic map of $$\mathbb {D}$$ . Suppose that f is in the Hardy space of the open unit disc $$H^p$$ . The operator that takes f into $$\psi \cdot f \circ \varphi $$ is a weighted composition operator, and is denoted by $$C_{\psi ,\varphi }$$ . The operator that takes f into $$\psi \cdot f^\prime \circ \varphi $$ is a weighted composition-differentiation operator. We prove that some weighted composition-differentiation operators belong to the closed algebra generated by weighted composition operators in the uniform operator topology.