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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 $$\sup_{| z| <1} (1-| z| ^2)^\alpha| f(z)| <\infty$$ and $$\int_{D}| f(z)| ^p\frac{\phi^p(| z| )}{1-| z| }\,dA(z)<\infty$$ 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.

MSC:
 47B38 Linear operators on function spaces (general) 47B33 Linear composition operators 30H05 Spaces of bounded analytic functions of one complex variable 46E15 Banach spaces of continuous, differentiable or analytic functions
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References:
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