## 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 $$\mathbb B^\alpha$$ and $$\mathbb B^\alpha_0$$, $$0 < \alpha < \infty$$, defined by $\mathbb B^\alpha = \{f \in H(D): \sup_{z\in D} (1 -| z| ^2)^\alpha | f'(z)| < \infty\}$ and $\mathbb B^\alpha_0 =\{f \in \mathbb B^\alpha : \lim_{| z| \to 1} (1 - | z| ^2)^\alpha | f'(z)| = 0\}.$ The main results completely characterize boundedness and compactness of $$uC_\phi$$ from $$\mathbb B^\alpha$$ to $$\mathbb B^\beta$$ as well as from $$\mathbb B^\alpha_0$$ to $$\mathbb B^\beta_0$$. 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 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 Linear composition operators 30D45 Normal functions of one complex variable, normal families 30H05 Spaces of bounded analytic functions of one complex variable

### Citations:

Zbl 0990.47018; Zbl 0959.47016
Full Text:

### References:

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