Weighted composition operators between Zygmund type spaces and their essential norms
type: Item
instance of: scholarly article
title: Weighted composition operators between Zygmund type spaces and their essential norms
zbMATH Open document ID: 1306.47036
author: Kobra Esmaeili
author: Mikael Lindström
published in: Integral Equations and Operator Theory
publication date: 2013-05-06T00:00:00Z
review text: Let \(H(\mathbb{D})\) denote the space of holomorphic functions on the unit disk \(\mathbb{D}\). For \(\alpha\in\mathbb{R}\), the space \(\Lambda^\alpha(\mathbb{D})\) consists of those functions \(f\in H(\mathbb{D})\) for which \[ |f^{(J)}(z)|(1-|z|)^{J-\alpha} \leq C, \quad z\in{\mathbb D}, \] where \(f^{(J)}\) is the derivative of order \(J\) and \(J\) is an integer such that \(J>\alpha\). In fact, \(\Lambda^\alpha(\mathbb{D})\), \(\alpha<0\), is a growth space, \(\Lambda^0(\mathbb{D})\) is the Bloch space, and \(\Lambda^\alpha(\mathbb{D})\), \(\alpha>0\), is the Lipschitz space of order \(\alpha\). In particular, \(\Lambda^1(\mathbb{D})\) is the Zygmund space. Given a function \(u\in H({\mathbb D})\) and a holomorphic mapping \(\varphi: {\mathbb D} \to {\mathbb D}\), the weighted composition operator \(uC_\varphi: H({\mathbb D}) \to H({\mathbb D})\) is defined by the formula \[ (uC_\varphi f)(z) = u(z) f(\varphi(z)), \quad f\in H({\mathbb D}),\quad z\in {\mathbb D}. \] For \(\alpha, \beta<2\), the authors characterize the bounded weighted composition operators \(uC_\varphi: \Lambda^\alpha({\mathbb D})\to \Lambda^\beta({\mathbb D})\). Also, they estimate the essential norms of the operators under consideration. Reviewer's remark. The bounded and compact weighted composition operators \(uC_\varphi: \Lambda^\alpha({\mathbb D})\to \Lambda^\beta({\mathbb D})\) were earlier characterized for all \(\alpha, \beta\in\mathbb{R}\) by the reviewer [J. Math. Sci., New York 182, No. 5, 630--638 (2012); translation from Zap. Nauchn. Semin. POMI 389, 85--100 (2011; Zbl 1257.30065)].
reviewed by: Evgueni Doubtsov
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 6160908
zbMATH Keywords: weighted composition operators
zbMATH Keywords: Lipschitz spaces
zbMATH Keywords: Zygmund spaces
zbMATH Keywords: essential norms
MaRDI profile type: Publication
full work available at URL: s00020-013-2038-4
OpenAlex ID: W2091183690
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cites work: EXISTENCE AND ASYMPTOTIC STABILITY OF SOLUTIONS OF A PERTURBED FRACTIONAL FUNCTIONAL-INTEGRAL EQUATION WITH LINEAR MODIFICATION OF THE ARGUMENT
cites work: Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions
cites work: Weighted composition operators between Bloch-type spaces.
cites work: Essential norms of weighted composition operators from the \(\alpha \)-Bloch space to a weighted-type space on the unit ball
cites work: Compact composition operators on BMOA and the Bloch space
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DOI: 10.1007/S00020-013-2038-4
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description: scientific article; zbMATH DE number 6160908