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Weighted composition operators between Bloch-type spaces. (English) Zbl 1042.47018

For analytic functions $u$ on the unit disk $D$ and analytic mappings $\varphi :D\to D$, the weighted composition operator $u{C}_{\varphi }$ is defined by $u{C}_{\varphi }\left(f\right)=u\left(f\circ \varphi \right)$ for $f$ analytic on $D$. In the paper under review, the authors consider these operators acting on the weighted Bloch-type spaces ${𝔹}^{\alpha }$ and ${𝔹}_{0}^{\alpha }$, $0<\alpha <\infty$, defined by

${𝔹}^{\alpha }=\left\{f\in H\left(D\right):\underset{z\in D}{sup}{\left(1-|z|}^{2}{\right)}^{\alpha }|{f}^{\text{'}}\left(z\right)|<\infty \right\}$

and

${𝔹}_{0}^{\alpha }=\left\{f\in {𝔹}^{\alpha }:\underset{|z|\to 1}{lim}{\left(1-|z|}^{2}{\right)}^{\alpha }|{f}^{\text{'}}\left(z\right)|=0\right\}·$

The main results completely characterize boundedness and compactness of $u{C}_{\varphi }$ from ${𝔹}^{\alpha }$ to ${𝔹}^{\beta }$ as well as from ${𝔹}_{0}^{\alpha }$ to ${𝔹}_{0}^{\beta }$. Finally, the authors give some examples of functions $u$ and $\varphi$ for which $u{C}_{\varphi }$ between the various spaces is bounded, compact or noncompact. Similar results were obtained by M. D. Contreras and A. G. Hernandez-Diaz [J. Aust. Math. Soc., Ser. A 69, 41–60 (2000; Zbl 0990.47018)] and A. Montes-Rodríguez [J. Lond. Math. Soc., II. Ser. 61, 872–884 (2000; Zbl 0959.47016)].

##### MSC:
 47B33 Composition operators 30D45 Bloch functions, normal functions, normal families 30H05 Bounded analytic functions