On a criterion for continuity and compactness of composition operators acting on \(\alpha\)-Bloch spaces
type: Item
instance of: scholarly article
title: On a criterion for continuity and compactness of composition operators acting on \(\alpha\)-Bloch spaces
zbMATH Open document ID: 1267.47043
author: María T. Malavé-Ramírez
author: Julio C. Ramos Fernández
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2013-03-15T00:00:00Z
review text: For \(\alpha>0\), the \(\alpha\)-Bloch space on the unit disc \(\mathbb{D}\), denoted by \(\mathcal{B}^{\alpha}\), is the collection of analytic functions \(f\) such that \[ \left\| f\right\|_\alpha=\sup_{z\in\mathbb{D}}\left(1-| z|^2\right)^{\alpha}\left| f'(z)\right|<\infty. \] For a self-map \(\phi\) of the disc, the composition operator \(C_\phi\) is defined by \(C_\phi f=f\circ \phi\). Let \(a\in\mathbb{D}\) and \(\sigma_{a}(z)=(1-| a|)\left(\left(1-\overline{a}z\right)^{-\alpha}-1\right)\). The authors prove that the composition operator \(C_\phi\) is bounded on \(\mathcal{B}^\alpha\) if and only if \[ \sup_{a\in\mathbb{D}} \left\| \sigma_a\circ\phi\right\|_{\alpha}<\infty. \] For compactness, the authors demonstrate that a vanishing version of the above condition is necessary and sufficient. These results are generalizations of previous work of \textit{M. Tjani} [Compact composition operators on some Moebius invariant Banach spaces. Ph.\,D.\ Thesis, Michigan State University (1996); Trans. Am. Math. Soc. 355, No.~11, 4683--4698 (2003; Zbl 1045.47020)].
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 30H30
zbMATH DE Number: 6145464
zbMATH Keywords: composition operators
zbMATH Keywords: Bloch space
reviewed by: Brett Wick
MaRDI profile type: Publication
full work available at URL: j.crma.2012.11.013
OpenAlex ID: W2058839707
cites work: Composition operators on \(\mu\)-Bloch type spaces
cites work: Compact Composition Operators on the Bloch Space
cites work: Q3144660
cites work: Compact composition operators on Besov spaces
cites work: Compactness of composition operators on BMOA and VMOA
cites work: Compact composition operators on BMOA and the Bloch space
cites work: Composition operators associated with bloch-type spaces
cites work: Essential norms of composition operators between Bloch type spaces
DOI: 10.1016/J.CRMA.2012.11.013
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description: scientific article; zbMATH DE number 6145464