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

Ohno, Shûichi; Stroethoff, Karel; Zhao, Ruhan

Rocky Mt. J. Math. 33, No. 1, 191-215 (2003).

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)].

Reviewer: Mikael Lindström (Åbo)

MSC:

47B33	Linear composition operators
30D45	Normal functions of one complex variable, normal families
30H05	Spaces of bounded analytic functions of one complex variable

Keywords:

weighted composition operator; weighted Bloch-type spaces; boundedness; compactness

Citations:

Zbl 0990.47018; Zbl 0959.47016

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