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Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk. (English) Zbl 1220.30072

Summary: We completely characterize the boundedness and compactness of composition operators from the space of Cauchy transforms on the unit disk $𝔻$ into the Bloch-type space ${ℬ}_{\nu }$, i.e., the space of all holomorphic functions $f$ on $𝔻$ such that

$\underset{z\in 𝔻}{sup}\nu \left(z\right)|{f}^{\text{'}}\left(z\right)|<\infty ,$

as well as into the little Bloch-type space ${ℬ}_{\nu ,0}$ consisting of all holomorphic functions $f$ on $𝔻$ such that

$\underset{|z|\to 1}{lim}\nu \left(z\right)|{f}^{\text{'}}\left(z\right)|=0$

for some weight function $\nu$. As a byproduct of our results, the norm of the operator is calculated when ${ℬ}_{\nu }$ is replaced by ${ℬ}_{\nu }/ℂ$.

##### MSC:
 30H99 Spaces and algebras of analytic functions
##### References:
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