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Composition followed by differentiation between Bloch type spaces. (English) Zbl 1132.47026
Let 𝔻 be the open unit disk in the complex plane. An analytic function f on 𝔻 is said to belong to α-Bloch space α if f α :=sup z𝔻 (1-|z| 2 ) α |f ' (z)|<. The little α-Bloch space 0 α is the subspace of α consisting of all f α for which (1-|z| 2 ) α |f ' (z)|0 as |z|1. These spaces are Banach spaces. Given an analytic self-map φ of 𝔻, let C φ denote the composition operator defined by C φ f=fφ for analytic functions f on 𝔻. Also, let D=/z be the complex differentiation operator. In this paper, the authors obtain characterizations for the boundedness and compactness of DC φ : α β . They also obtain a characterization for the compactness of DC φ : α 0 β .
MSC:
47B38Operators on function spaces (general)
30D45Bloch functions, normal functions, normal families
30H05Bounded analytic functions
47B33Composition operators