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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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