Weighted composition operators from the minimal Möbius invariant space into the Bloch space
type: Item
instance of: scholarly article
title: Weighted composition operators from the minimal Möbius invariant space into the Bloch space
zbMATH Open document ID: 1302.47036
author: Flavia Colonna
author: Songxiao Li
published in: Mediterranean Journal of Mathematics
publication date: 2013-03-20T00:00:00Z
review text: The authors study weighted composition operators \(uC_\varphi (f)= u(z)f(\varphi (z))\) for analytic functions \(u\) and analytic self-maps \(\varphi\) of the unit disc acting from the minimal Möbius invariant space, corresponding to the Besov space \(B_1\), into the maximal Möbius invariant space, corresponding to the Bloch space \(\mathcal B\). It is shown that boundedness can be described by the condition \(\sup_n \|u\varphi^n\|_{\mathcal B}<\infty\) or by the conditions \(u\in \mathcal B\) together with \(\sup_{|z|<1}\frac{(1-|z|^2)|u(z)\varphi'(z)|}{1-|\varphi (z)|^2}<\infty\). Regarding the compactness of \(uC_{\varphi}\), it is shown to be equivalent, among other things, to \(\|g_n\|_{\mathcal B}+\|u\varphi^n\|_{\mathcal B} \to 0\) as \(n\to \infty\), where \(g_n(z)=\int_0^z u'(\xi)\varphi(\xi)^n\,d\xi\).
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 30H30
zbMATH DE Number: 6147195
zbMATH Keywords: weighted composition operator
zbMATH Keywords: minimal Möbius invariant space
zbMATH Keywords: Bloch space
MaRDI profile type: Publication
full work available at URL: s00009-012-0182-8
OpenAlex ID: W2046336509
cites work: Q3681112
cites work: Q3505627
cites work: New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space
cites work: Weighted composition operators from the Besov spaces into the Bloch spaces
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 \(H^{\infty }\) to the Bloch space on the polydisc
cites work: Weighted composition operators from Bergman-type spaces into Bloch spaces
cites work: Weighted composition operators between \(H^\infty\) and \(\alpha\)-Bloch spaces in the unit ball
cites work: Weighted composition operators from Zygmund spaces into Bloch spaces
cites work: Composition operator on Bloch type spaces
cites work: Essential norms of weighted composition operators between Bloch-type spaces
cites work: Compact Composition Operators on the Bloch Space
cites work: The essential norm of a composition operator on Bloch spaces
cites work: Weighted composition operators between \(H^\infty\) and the Bloch space
cites work: Weighted composition operators between Bloch-type spaces.
cites work: An Extremal Property of the Bloch Space
cites work: Composition operators and classical function theory
cites work: Essential norms of weighted composition operators from the \(\alpha \)-Bloch space to a weighted-type space on the unit ball
cites work: Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball
cites work: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball
cites work: On an integral operator between Bloch-type spaces on the unit ball
cites work: Compact composition operators on Besov spaces
cites work: Q4932109
cites work: Compact composition operators on BMOA and the Bloch space
cites work: Norm of composition operators on the Bloch space
cites work: Q3011809
cites work: Composition operators from Bloch type spaces to Hardy and Besov spaces
cites work: Essential norms of composition operators between Bloch type spaces
cites work: Q3999007
cites work: Weighted composition operators from \(F(p,q,s)\) spaces to \(H_{\mu }^{\infty}\) spaces
cites work: Weighted composition operators from area Nevanlinna spaces into Bloch spaces
DOI: 10.1007/S00009-012-0182-8
description: scientific article; zbMATH DE number 6147195