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Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces. (English) Zbl 1119.47035
Let H α and A p (ϕ) denote the spaces of analytic functions in the unit disk such that sup |z|<1 (1-|z| 2 ) α |f(z)|< and D |f(z)| p ϕ p (|z|) 1-|z|dA(z)< for a normal weight function ϕ, 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 φ,u n f=uf (n) φ, where u is a given analytic function and φ is a non-constant analytic self-map on the disk, and between the spaces A p (ϕ) and H α . Due to the fact that D φ,u n gives DC φ , C φ D and M u D as particular cases, his results allow him to unify a number of previously known theorems.

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