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Generalized weighted composition operators on weighted Bergman spaces. (English) Zbl 1183.47030
Let $\mathbb{D}$ be the open unit disk in the complex plane and $H(\mathbb{D})$ be the set of all analytic functions on $\mathbb{D}$. An analytic self-map $\varphi$ of $\mathbb{D}$ induces a linear composition operator $C_{\varphi}$ defined by $$ C_{\varphi}: H(\mathbb{D}) \to H(\mathbb{D}), \; f \mapsto f \circ \varphi. $$ Moreover, let $D$ be the differentiation operator. We consider $$ D: H(\mathbb{D}) \to(\mathbb{D}), \; f \mapsto f' \text{ and } D^n: H(\mathbb{D}) \to(\mathbb{D}), \; f \mapsto f^{(n)} \text{ for every } n \in \mathbb{N}, \; n \ge 2. $$ In this paper, the author studies the operator $D_{\varphi, \psi}^n$ defined by $$D_{\varphi, \psi}^n f= \psi f^{(n)} \circ \varphi, \; f \in H(\mathbb{D}),$$ acting between the weighted Bergman spaces $$A_{\alpha}^p= \left \{ f \in H(\mathbb{D})\mid \|f\|_{A_{\alpha}^p}^p= \int_{\mathbb{D}} |f(z)|^p \; d A_{\alpha}(z) < \infty \right \},$$ where $dA_{\alpha}(z)= (\alpha +1) (1-|z|^2)^{\alpha} dA$, $dA$ is the normalized Lebesgue measure, $0<p< \infty$ as well as $-1 < \alpha < \infty$. Boundedness and compactness of such operators acting between different weighted Bergman spaces as well as between a weighted Bergman space and the space $H^{\infty}$ of all bounded analytic functions on $\mathbb{D}$ are characterized.

47B38Operators on function spaces (general)
30H20Bergman spaces, Fock spaces
47B33Composition operators
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