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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 $\phi: D \to D$, the weighted composition operator $uC_\phi$ is defined by $uC_\phi(f) = u(f \circ \phi)$ for $f$ analytic on $D$. In the paper under review, the authors consider these operators acting on the weighted Bloch-type spaces $\Bbb B^\alpha$ and $\Bbb B^\alpha_0$, $ 0 < \alpha < \infty$, defined by $$\Bbb B^\alpha = \{f \in H(D): \sup_{z\in D} (1 -\vert z\vert ^2)^\alpha \vert f'(z)\vert < \infty\}$$ and $$\Bbb B^\alpha_0 =\{f \in \Bbb B^\alpha : \lim_{\vert z\vert \to 1} (1 - \vert z\vert ^2)^\alpha \vert f'(z)\vert = 0\}.$$ The main results completely characterize boundedness and compactness of $uC_\phi$ from $\Bbb B^\alpha$ to $\Bbb B^\beta$ as well as from $\Bbb B^\alpha_0$ to $\Bbb B^\beta_0$. Finally, the authors give some examples of functions $u$ and $\phi$ for which $uC_\phi$ between the various spaces is bounded, compact or noncompact. Similar results were obtained by {\it M. D. Contreras} and {\it A. G. Hernandez-Diaz} [J. Aust. Math. Soc., Ser. A 69, 41--60 (2000; Zbl 0990.47018)] and {\it A. Montes-Rodríguez} [J. Lond. Math. Soc., II. Ser. 61, 872--884 (2000; Zbl 0959.47016)].

MSC:
47B33Composition operators
30D45Bloch functions, normal functions, normal families
30H05Bounded analytic functions
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References:
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