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

47B38Operators on function spaces (general)
30D45Bloch functions, normal functions, normal families
30H05Bounded analytic functions
47B33Composition operators