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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.

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
Full Text: DOI
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