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\begin{document}$$\varphi $$\end{document} be an analytic self map of the open unit disc D\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {D}$$\end{document}. Assume that ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} is an analytic map of D\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {D}$$\end{document}. Suppose that f is in the Hardy space of the open unit disc Hp\documentclass[12pt]{minimal}
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\begin{document}$$H^p$$\end{document}. The operator that takes f into ψ·f∘φ\documentclass[12pt]{minimal}
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\begin{document}$$\psi \cdot f \circ \varphi $$\end{document} is a weighted composition operator, and is denoted by Cψ,φ\documentclass[12pt]{minimal}
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\begin{document}$$C_{\psi ,\varphi }$$\end{document}. The operator that takes f into ψ·f′∘φ\documentclass[12pt]{minimal}
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\begin{document}$$\psi \cdot f^\prime \circ \varphi $$\end{document} 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.
