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Products of Volterra type operator and composition operator from H and Bloch spaces to Zygmund spaces. (English) Zbl 1145.47022
The Zygmund space 𝒵 is the set of all analytic functions f on the unit disc 𝔻 such that f 𝒵 =|f(0)|+|f ' (0)|+sup z (1-|z| 2 )|f '' (z)|<+, endowed with such a norm. If lim |z|1 (1-|z| 2 )|f '' (z)|=0, then f is said to belong to the little Zygmund space 𝒵 0 · Given an analytic function gH(𝔻), two types of Volterra integral operator are defined according to J g (f)(z)= 0 z fg ' and I g (f)(z)= 0 z f ' g for fH(𝔻)· The authors consider 𝒵 and 𝒵 0 valued compositions of these operators and composition operators C φ whose symbol φ is an analytic self-map of 𝔻· They compare their boundedness or compactness regarding the domain space, specifically, H , the Bloch space , or the little Bloch space 0 · As a sample of the paper results, let us quote the following (Theorem 1): Set T=C φ I g · Then T:H 𝒵 is bounded if and only if T:𝒵 is bounded if and only if T: 0 𝒵 is bounded. Similarly, for the compact case.

47B33Composition operators
47B38Operators on function spaces (general)
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