Products of differentiation and composition operators on the Bloch space
type: Item
instance of: scholarly article
title: Products of differentiation and composition operators on the Bloch space
zbMATH Open document ID: 1267.30087
author: Hasi Wulan
author: Yu Tian Wu
published in: Collectanea Mathematica
publication date: 2012-12-04T00:00:00Z
review text: Let \(H({\mathbb{D}})\) be the space of all analytic functions defined on the unit disk \({\mathbb{D}}\) in the complex plane \(\mathbb C\). Let \(\sigma_a(z)=\frac{a-z}{1-\bar{a}z}\) be the Möbius transformation on \({\mathbb{D}}\). For a nonconstant analytic self-map \(\phi\) of \({\mathbb{D}}\), the composition operator \(C_\phi\) on \(H({\mathbb{D}})\) is defined by \(C_\phi f=f\circ \phi\). In the paper under review the authors consider linear product operators \[ DC_\phi f=(f\circ\phi)'\quad\text{and}\quad C_\phi D^m f=f^{(m)}\circ \phi \] acting on the well known Bloch space \(\mathcal{B}\), where \(m\in \mathbb{Z}\). The authors give new and simple compactness criteria for the operator \(DC_\phi\) on \(\mathcal{B}\). The paper contains six new theorems, three lemmas and one interesting example. Theorem 2.1 says that \(DC_\phi\) is compact on \(\mathcal{B}\) if and only if \(DC_\phi\) is bounded on \(\mathcal{B}\) and \[ \lim_{n\to\infty}\|D\phi ^n\|_\mathcal{B} =0. \] In the next section they prove that \(C_\phi D^m\) is compact on \(\mathcal{B}\) if and only if \[ \lim_{|a|\to 1} \|C_\phi D^m\sigma_a\|_\mathcal{B} = 0. \]
reviewed by: Janusz Godula
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30H30
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 6112444
zbMATH Keywords: composition operator
zbMATH Keywords: differentiation operator
zbMATH Keywords: Bloch space
MaRDI profile type: Publication
full work available at URL: s13348-010-0030-8
OpenAlex ID: W2093324071
cites work: Q4888816
cites work: Composition followed by differentiation between Bergman and Hardy spaces
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cites work: Q3622002
cites work: Weighted composition operators between \(H^\infty\) and the Bloch space
cites work: Products of composition and differentiation between Hardy spaces
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cites work: Compact composition operators on BMOA and the Bloch space
cites work: Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces
DOI: 10.1007/S13348-010-0030-8
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description: scientific article; zbMATH DE number 6112444