A note on weighted composition operators on the weighted Bergman space
type: Item
instance of: scholarly article
title: A note on weighted composition operators on the weighted Bergman space
zbMATH Open document ID: 1339.47038
author: Liankuo Zhao
author: Shengzhao Hou
publication date: 2015-10-26T00:00:00Z
review text: For the unit disk \(\mathbb{D}\) and the space \(H(\mathbb{D})\) of all analytic functions on \(\mathbb{D}\), \(1 < p < \infty\), \(\alpha > -1\), the weighted Bergman space \({\mathcal{L}}_a^p(dA_{\alpha})\) is defined as \({\mathcal{L}}_a^p(dA_{\alpha}) = H(\mathbb{D}) \cap {\mathcal{L}}^p(\mathbb{D},dA_{\alpha})\), where \(dA_{\alpha}(z)=(\alpha+1)(1-|z|^2)^{\alpha}dA(z)\) is the Möbius invariant measure, and \(dA\) denotes the normalized area measure on \(\mathbb{D}\). For an analytic self-map \(\varphi\) on \(\mathbb{D}\) and \(\psi \in H(\mathbb{D})\), the weighted composition operator \(C_{\psi,\varphi}\) on \(H(\mathbb{D})\) is defined as \(C_{\psi,\varphi}=M_{\psi}\circ C_{\varphi} : f \mapsto \psi\cdot (f\circ\varphi)\) for \(f \in H(\mathbb{D})\), where \(M_{\psi}\) and \(C_{\varphi}\) are the constituent multiplication and composition operators with the respective symbols \(\psi\) (weighted symbol) and \(\varphi\) (composition symbol). An analytic self-map \(\varphi\) is a finite Blaschke product if \[ \varphi =z^m \prod\limits_{k=1}^n\frac{\overline{a}_k(a_k-z)}{a_k(1-\overline{a}_kz)}, \] where \(a_k \in \mathbb{D} \setminus\{0\}\), \(k=1,2,\dots{},n\). It is well known that \(C_{\varphi}\) is bounded on \({\mathcal{L}}_a^p(dA_{\alpha})\) for any self-map \(\varphi\) of \(\mathbb{D}\) and \(M_{\psi}\) is bounded on \({\mathcal{L}}_a^p(dA_{\alpha})\) if and only if \(\psi \in H^{\infty}(\mathbb{D})\), the space of all bounded analytic functions on \(\mathbb{D}\). Clearly, bounded multiplicative and composition operators yield a bounded weighted composition operator \(C_{\psi,\varphi}\). However, the converse is not true, as shown in [\textit{M. D. Contreras} and \textit{A. G. Hernández-Díaz}, Integral Equations Oper. Theory 46, No. 2, 165--188 (2003; Zbl 1042.47017)]. For the Hardy space \(H^p\), \(1 < p < \infty\), it was shown in [loc. cit.] that \(\{ \psi \in H^p : C_{\psi,\varphi}H^p \subset H^p \} = H^{\infty}(\mathbb{D})\) if and only if \(\varphi…
reviewed by: Abebaw Tadesse
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 6498080
zbMATH Keywords: weighted Bergman space
zbMATH Keywords: weighted composition operator
zbMATH Keywords: boundedness
zbMATH Keywords: finite Blaschke product
MaRDI profile type: Publication
full work available at URL: s10114-015-4473-4
OpenAlex ID: W2172430116
cites work: Weighted composition operators between different Hardy spaces
cites work: Q4515420
cites work: WEIGHTED COMPOSITION OPERATORS ON THE BERGMAN SPACE
cites work: Weighted composition operators between different weighted Bergman spaces and different Hardy spaces
cites work: Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions
cites work: Q3016202
cites work: Weighted composition operators between Bloch-type spaces.
cites work: Q5567211
cites work: Weighted composition operators between weighted Bergman spaces
DOI: 10.1007/S10114-015-4473-4
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 6498080