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Composition followed by differentiation between Bloch type spaces. (English) Zbl 1132.47026
Let $$\mathbb D$$ be the open unit disk in the complex plane. An analytic function $$f$$ on $$\mathbb D$$ is said to belong to $$\alpha$$-Bloch space $$\mathcal B^\alpha$$ if $$\| f\| _{\mathcal B_\alpha}:=\sup_{z\in\mathbb D} (1-| z| ^2)^\alpha| f'(z)| <\infty$$. The little $$\alpha$$-Bloch space $${\mathcal B}^\alpha_0$$ is the subspace of $$\mathcal B^\alpha$$ consisting of all $$f\in\mathcal B^\alpha$$ for which $$(1-| z| ^2)^\alpha| f'(z)| \to 0$$ as $$| z| \to 1$$. These spaces are Banach spaces. Given an analytic self-map $$\varphi$$ of $$\mathbb D$$, let $$C_\varphi$$ denote the composition operator defined by $$C_\varphi f= f\circ \varphi$$ for analytic functions $$f$$ on $$\mathbb D$$. Also, let $$D=\partial/\partial z$$ be the complex differentiation operator. In this paper, the authors obtain characterizations for the boundedness and compactness of $$DC_\varphi:{\mathcal B}^\alpha\to{\mathcal B}^\beta$$. They also obtain a characterization for the compactness of $$DC_\varphi:{\mathcal B}^\alpha\to{\mathcal B}_{0}^\beta$$.

##### MSC:
 47B38 Linear operators on function spaces (general) 30D45 Normal functions of one complex variable, normal families 30H05 Spaces of bounded analytic functions of one complex variable 47B33 Linear composition operators