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The composition operators on weighted Bloch space. (English) Zbl 1038.47020

Let $H\left(D\right)$ denote the set of all holomorphic functions in the open unit disk $D$ and let $\varphi$ be a holomorphic selfmap of $D$. Suppose that $X$ and $Y$ are two function spaces defined on $D$. The map $\varphi$ is said to have the $X$-to-$Y$ pullback property if $f\circ \varphi \in Y$ for every $f\in X$. Let $I$ be an arc on the unit circle and $S\left(I\right)=\left\{r{e}^{i\theta }:1-r\le |I|$, ${e}^{i\theta }\in I\right\}$ the associated Carleson box. In the present paper, the author studies the $X$-to-$Y$ pullback property for

$X=Y={B}_{log}:=\left\{f\in H\left(D\right)\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}\underset{z\in D}{sup}{\left(1-|z|}^{2}\right)\left(log\frac{2}{1-{|z|}^{2}}\right)|{f}^{\text{'}}\left(z\right)|<\infty \right\},$

for

$X=Y={\text{BMOA}}_{log}:=\left\{f\in H\left(D\right)\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}\underset{I}{sup}\frac{{\left(log\frac{2}{|I|}\right)}^{2}}{|I|}{\int }_{S\left(I\right)}|{f}^{\text{'}}{\left(z\right)|}^{2}log\left(1/|z|\right)dA\left(z\right)<\infty \right\},$

and for $X={B}_{log}$ and $Y={\text{BMOA}}_{log}$. A complete characterization of the bounded and compact composition operators ${C}_{\varphi }:f↦f\circ \varphi$ on the weighted Bloch space ${B}_{log}$ is given. Unfortunately, this paper has many grammatical errors and the title of reference [1] is not correct.

##### MSC:
 47B33 Composition operators 30H05 Bounded analytic functions