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Products of Volterra type operator and composition operator from ${H}^{\infty }$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 ${\parallel f\parallel }_{𝒵}=|f\left(0\right)|+|{f}^{\text{'}}\left(0\right)|+{sup}_{z}{\left(1-|z|}^{2}\right)|{f}^{\text{'}\text{'}}\left(z\right)|<+\infty ,$ endowed with such a norm. If ${lim}_{|z|\to 1}{\left(1-|z|}^{2}\right)|{f}^{\text{'}\text{'}}\left(z\right)|=0,$ then $f$ is said to belong to the little Zygmund space ${𝒵}_{0}·$ Given an analytic function $g\in H\left(𝔻\right),$ two types of Volterra integral operator are defined according to ${J}_{g}\left(f\right)\left(z\right)={\int }_{0}^{z}f{g}^{\text{'}}$ and ${I}_{g}\left(f\right)\left(z\right)={\int }_{0}^{z}{f}^{\text{'}}g$ for $f\in H\left(𝔻\right)·$ The authors consider $𝒵$ and ${𝒵}_{0}$ valued compositions of these operators and composition operators ${C}_{\varphi }$ whose symbol $\varphi$ is an analytic self-map of $𝔻·$ They compare their boundedness or compactness regarding the domain space, specifically, ${H}^{\infty },$ the Bloch space $ℬ,$ or the little Bloch space ${ℬ}_{0}·$ As a sample of the paper results, let us quote the following (Theorem 1): Set $T={C}_{\varphi }\circ {I}_{g}·$ Then $T:{H}^{\infty }\to 𝒵$ is bounded if and only if $T:ℬ\to 𝒵$ is bounded if and only if $T:{ℬ}_{0}\to 𝒵$ is bounded. Similarly, for the compact case.
##### MSC:
 47B33 Composition operators 47B38 Operators on function spaces (general)