A note on spectra of weighted composition operators on weighted Banach spaces of holomorphic functions
type: Item
instance of: scholarly article
title: A note on spectra of weighted composition operators on weighted Banach spaces of holomorphic functions
zbMATH Open document ID: 1159.47010
author: Elke Wolf
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2008-11-27T00:00:00Z
full work available at URL: https://eudml.org/doc/55283
full work available at URL: v31n2p4.pdf
review text: The author shows that, if \(v\) is a radial weight in the unit disc, nonincreasing in \(| z| \) and \(\lim_{| z| \to \infty}v(z)=0\), \(\phi:D\to D\) is an analytic self map which is not an automorphism and has a fixed point \(a\in D\), and \(\psi\) is analytic in \(D\) satisfying that \(C_{\phi,\psi}(f)=\psi (f\circ \phi)\) is bounded on \(H^\infty_v\), then the spectrum of the weighted composition operator \(C_{\phi,\psi}\) is given by the ball centered at the origin and radius the essential spectral radius together with the sequence \(\{\psi(a) \phi'(a)^n\}\). The proof follows the ideas from the previous works in the cases \(\psi(z)=1\) by \textit{R.\,Aron} and \textit{M.\,Lindström} for \(v(z)= (1-| z| ^2)^p\) [Isr.\ J.\ Math.\ 141, 263--276 (2004; Zbl 1074.47010)] and by \textit{J.\,Bonet, P.\,Galindo} and \textit{M.\,Lindström} for general weights [J.~Math.\ Anal.\ Appl.\ 340, No.\,2, 884--891 (2008; Zbl 1138.47017)].
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5374767
zbMATH Keywords: spectra of weighted composition operators
zbMATH Keywords: weighted Banach spaces of analytic functions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5374767