Chaotic weighted shifts in Bargmann space
type: Item
instance of: scholarly article
title: Chaotic weighted shifts in Bargmann space
zbMATH Open document ID: 1072.47007
author: G. S. Heshmati
author: Hassan Emamirad
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-06-30T00:00:00Z
review text: The authors study the chaotic behaviour of unbounded weighted backward shifts on the Bargmann space \(F(\mathbb{C})\), that is, the set of entire functions which can be written as \(f(z)=\sum_{k\geq 0} c_{k} \frac {z^{k}}{\sqrt{k!}}\), where \(\sum_{k\geq 0} | c_{k}| ^{2}\) is finite. If \(T\) is the backward shift with weights \((\omega _{k})_{k\geq 0}\) defined by \[ T\left(\sum_{k\geq 0} c_{k} \frac {z^{k}}{\sqrt{k!}}\right)= \sum_{k\geq 0} \omega _{k}c_{k+1} \frac {z^{k}}{\sqrt{k!}}, \] it is proved that \(T\) is chaotic if and only if the series \[ \sum_{n=1}^{\infty } \prod_{j=0}^{n-1}\frac {1}{| \omega _{j}| ^{2}} \] is convergent. In particular, the annihilation operator \(A: f \longmapsto f'\) is chaotic on \(F(\mathbb{C})\).
reviewed by: Sophie Grivaux
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 2184068
zbMATH Keywords: backward weighted shifts
zbMATH Keywords: Bargmann space
zbMATH Keywords: hypercyclic operators
zbMATH Keywords: chaotic operators
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.10.035
OpenAlex ID: W1985407051
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DOI: 10.1016/J.JMAA.2004.10.035
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description: scientific article; zbMATH DE number 2184068