Univalent symbols of Volterra operators on growth spaces
type: Item
instance of: scholarly article
title: Univalent symbols of Volterra operators on growth spaces
zbMATH Open document ID: 1435.30154
author: Evgeny Abakumov
author: Evgueni Doubtsov
published in: Analysis and Mathematical Physics
publication date: 2019-11-28T00:00:00Z
review text: Starting from the fact that the Voltera operator \(T_g:H^\infty\rightarrow H^\infty\) is bounded if \(g\) is a univalent function, this paper extends this result to the case when \(T_g\) is an operator on arbitrary growth spaces. The growth space \(\mathcal{ A}^w(\mathbb D)\) consists of the holomorphic functions \(f\) on the unit disc \(\mathbb D\) for which \(\|f\|_{\mathcal{ A}^w}=\sup\frac{|f(z)|}{w(z)}<\infty\), where \(w:[0,1)\rightarrow(0,+\infty)\) is a weight function. For a weight function \(w\) the associated weight \(\tilde{w}\) is defined as \[\tilde{w}(z)=\sup\big\{|f(z)|: f \text{ holomorphic on }\mathbb D\text{ and }\vert f\vert\le w\text{ on }\mathbb D\big\}.\] The main result of the paper (Theorem 1) gives a characterization of those univalent functions \(g\) for which the Voltera operator \(T_g:\mathcal{ A}^w(\mathbb D)\rightarrow \mathcal{ A}^v(\mathbb D)\) is bounded, in terms of the associated weight functions. Also, a similar characterization of \(T_g:\mathcal{ A}^w(\mathbb D)\rightarrow \mathcal{ A}^v(\mathbb D)\) to be compact when \(g\) is a univalent function is derived.
reviewed by: Ilie Valuşescu
Mathematics Subject Classification ID: 30H99
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 7138520
zbMATH Keywords: spaces of analytic functions
zbMATH Keywords: Volterra operators
MaRDI profile type: Publication
full work available at URL: s13324-019-00283-0
OpenAlex ID: W2914947001
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Wikidata QID: Q128424572
DOI: 10.1007/S13324-019-00283-0
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description: scientific article; zbMATH DE number 7138520