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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 \(\mathbb B^\alpha\) 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
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References:

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