Norm of weighted composition operators from \(\alpha \)-Bloch spaces to weighted-type spaces
type: Item
instance of: scholarly article
title: Norm of weighted composition operators from \(\alpha \)-Bloch spaces to weighted-type spaces
zbMATH Open document ID: 1181.32011
author: Stevo Stević
published in: Applied Mathematics and Computation
publication date: 2009-10-09T00:00:00Z
review text: For \(\alpha>0\), let \(\mathcal B^\alpha\) be the \(\alpha\)-Bloch space in the unit ball \(\mathbb B_n\) of \(\mathbb C^n\). The author endows this space with the norm \[ \|f\|_{\mathcal B^\alpha}:=|f(0)|+\sup_{z\in \mathbb B^n} (1-|z|)^\alpha |\nabla f(z)|. \] Also, for a positive continuous function \(\mu\) on \(\mathbb B^n\), called a weight, let \(H^\infty_\mu\) be the space of all \(f\) holomorphic in \(\mathbb B^n\) for which \(\|f\|_{H^\infty_\mu}:=\sup_{z\in \mathbb B^n} \mu(z)|f(z)|<\infty\). The paper contains a single result: the explicite determination of the norm of the weighted composition operator \(T=uC_\varphi: \mathcal B^\alpha\to H^\infty_\mu\), \(f\mapsto u(f\circ \varphi)\), where \(u:\mathbb B_n\to \mathbb C\) is a holomorphic function and \(\varphi\) a holomorphic selfmap of \(\mathbb B_n\). It is shown that, if \(\alpha\not=1\) and \(T\) is bounded, then \[ \|T\|_{\text{op}}=\max\left\{ \|u\|_{H^\infty_\mu} , \sup_{z\in\mathbb B_n} \frac{\mu(z)|u(z)|}{\alpha-1} \left( \frac{1}{(1-|\varphi(z)|)^{\alpha-1}}-1 \right) \right\}. \] The case \(\alpha=1\) was done in [\textit{S. Stevic}, Ars. Combin. 88, 125--127 (2008)].
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 32A18
zbMATH DE Number: 5612859
zbMATH Keywords: weighted composition operator
zbMATH Keywords: Bloch space
zbMATH Keywords: weighted-type space
zbMATH Keywords: operator norm
MaRDI profile type: Publication
full work available at URL: j.amc.2009.06.005
OpenAlex ID: W2029267977
cites work: Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball
cites work: Q4888816
cites work: Weighted composition operators on some weighted spaces in the unit ball
cites work: Weighted composition operators from generalized weighted Bergman spaces to weighted-type spaces
cites work: Weighted composition operators from Bergman-type spaces into Bloch spaces
cites work: Weighted Composition Operators from α-Bloch Space to H∞on the Polydisc
cites work: Weighted composition operators from \(H^{\infty }\) to the Bloch space on the polydisc
cites work: Weighted composition operators between \(H^\infty\) and \(\alpha\)-Bloch spaces in the unit ball
cites work: Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions
cites work: Weighted composition operators between \(H^\infty\) and the Bloch space
cites work: On an integral operator on the unit ball in \(\mathbb{C}^n\)
cites work: Composition operators between \(H^\infty\) and \(a\)-Bloch spaces on the polydisc
cites work: Weighted composition operators between mixed norm spaces and \(H_{\alpha }^{\infty }\) spaces in the unit ball
cites work: Essential norms of weighted composition operators from the \(\alpha \)-Bloch space to a weighted-type space on the unit ball
cites work: Q3534271
cites work: Q3070312
cites work: Norm and essential norm of composition followed by differentiation from \(\alpha \)-Bloch spaces to \(H_\mu^\infty\)
cites work: Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball
cites work: Weighted composition operators on some function spaces of entire functions
cites work: Compact weighted composition operators and multiplication operators between Hardy spaces
cites work: Q3653739
cites work: Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions
cites work: Norm of composition operators on the Bloch space
cites work: Q3011809
DOI: 10.1016/J.AMC.2009.06.005
description: scientific article; zbMATH DE number 5612859