×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] J. Arazy, Multipliers of Bloch functions , University of Haifa Publication Series 54 , 1982.
[2] L. Brown and A.L. Shields, Multipliers and cyclic vectors in the Bloch space , Michigan Math. J. 38 (1991), 141-146. · Zbl 0749.30023 · doi:10.1307/mmj/1029004269
[3] P.L. Duren, Theory of \(H^p\) spaces , Academic Press, New York, 1970. · Zbl 0215.20203
[4] G.H. Hardy and J.E. Littlewood, Some properties of fractional integrals II, Math. Z. 34 (1932), 403-439. · Zbl 0003.15601 · doi:10.1007/BF01180596 · eudml:168318
[5] K.M. Madigan, Composition operators on analytic Lipschitz spaces , Proc. Amer. Math. Soc. 119 (1993), 465-473. JSTOR: · Zbl 0793.47037 · doi:10.2307/2159930 · links.jstor.org
[6] K.M. Madigan and A. Matheson, Compact composition operators on the Bloch space , Trans. Amer. Math. Soc. 347 (1995), 2679-2687. JSTOR: · Zbl 0826.47023 · doi:10.2307/2154848 · links.jstor.org
[7] S. Ohno and R. Zhao, Weighted composition operators on the Bloch space , Bull. Austral. Math. Soc., 63 (2001), 177-185. · Zbl 0985.47022 · doi:10.1017/S0004972700019250
[8] R.C. Roan, Composition operators on a space of Lipschitz functions , Rocky Mountain J. Math. 10 (1980), 371-379. · Zbl 0433.46023 · doi:10.1216/RMJ-1980-10-2-371
[9] J.H. Shapiro, Composition operators and classical function theory , Springer-Verlag, New York, 1993. · Zbl 0791.30033
[10] J. Xiao, Composition operators associated with Block-type spaces , Complex Variables Theory Appl. 46 (2001), 109-121. · Zbl 1044.47020
[11] K.H. Zhu, Operator theory on function spaces , Marcel Dekker, New York, 1990. · Zbl 0706.47019
[12] ——–, Bloch type spaces of analytic functions , Rocky Mountain J. Math. 23 (1993), 1143-1177. · Zbl 0787.30019 · doi:10.1216/rmjm/1181072549
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.