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Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces. (English) Zbl 1119.47035
Let $H^\infty_\alpha$ and $A^p(\phi)$ denote the spaces of analytic functions in the unit disk such that $\sup_{\vert z\vert <1} (1-\vert z\vert ^2)^\alpha\vert f(z)\vert <\infty$ and $\int_{D}\vert f(z)\vert ^p\frac{\phi^p(\vert z\vert )}{1-\vert z\vert }\,dA(z)<\infty$ for a normal weight function $\phi$, respectively. The author completely characterizes the boundedness and compactness of the composition between differentiation and multiplication operator $DM_u$ and also for the operators $D^n_{\varphi,u} f = u f^{(n)}\circ \varphi$, where $u$ is a given analytic function and $\varphi$ is a non-constant analytic self-map on the disk, and between the spaces $A^p(\phi)$ and $H^\infty_\alpha$. Due to the fact that $D^n_{\varphi,u}$ gives $DC_\varphi$, $C_\varphi D$ and $M_uD$ as particular cases, his results allow him to unify a number of previously known theorems.

47B38Operators on function spaces (general)
47B33Composition operators
30H05Bounded analytic functions
46E15Banach spaces of continuous, differentiable or analytic functions
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