Orbits of semigroups of truncated convolution operators
type: Item
instance of: scholarly article
zbMATH Open document ID: 1243.47023
DOI: 10.1017/S0017089512000055
published in: Glasgow Mathematical Journal
publication date: 2012-05-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.3484
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 37B99
zbMATH DE Number: 6031021
zbMATH Keywords: universal families
zbMATH Keywords: universal elements
zbMATH Keywords: supercyclic families
zbMATH Keywords: hypercyclic families
zbMATH Keywords: Volterra operators
zbMATH Keywords: truncated convolution operators
zbMATH Keywords: irregular vectors
author: S. Shkarin
MaRDI profile type: Publication
arXiv ID: 1008.3484
cites work: The Volterra operator is not supercyclic
cites work: Hypercyclic tuples of operators and somewhere dense orbits
cites work: Non-supercyclicity of Volterra convolution and related operators
cites work: Orbits of operators commuting with the Volterra operator
cites work: Q3994587
cites work: Operators commuting with the Volterra operator are not weakly supercyclic
cites work: ANTISUPERCYCLIC OPERATORS AND ORBITS OF THE VOLTERRA OPERATOR
cites work: Irregular vectors of Hilbert space operators
title: Orbits of semigroups of truncated convolution operators
review text: A family \({\mathcal F} = \{ F_a : a \in A \}\) of continuous maps from a topological space \(X\) to a topological space \(Y\) is called \textit{universal} if there exists \(x \in X\) such that the orbit \({\mathcal O}({\mathcal F}, x) =\{ F_a x : a \in A \}\) is dense in \(Y\). The element \(x\) is called an \textit{universal element} for the family \({\mathcal F}\), the set of all such elements is denoted by \({\mathcal U}({\mathcal F})\). This concept is applied in the particular case where \(X = Y\) is a topological vector space, the maps belong to the set \(L(X)\) of continuous linear operators from \(X\) into itself and the family \(\mathcal F\) is a commutative subsemigroup of \(L(X)\), in which case the word \textit{universal} is replaced by \textit{hypercyclic}. Finally, \(\mathcal F\) is \textit{supercyclic} if the semigroup \({\mathcal F}_p = \{zT : T \in {\mathcal F},\;z\text{ a scalar}\}\) is hypercyclic; hypercyclic vectors for \({\mathcal F}_p\) are called \textit{supercyclic vectors} for \({\mathcal F}\). A~finite set \({\mathbf T} = \{ T_1, T_2, \dots, T_n\}\) of commuting operators in \(L(X)\) is hypercyclic (supercyclic) if the semigroup generated by \textbf{T} enjoys the corresponding property.NEWLINENEWLINEIt is known that, if \(1 \leq p < \infty\), the Volterra operator NEWLINE\[NEWLINE Vf(x) = \int_0^x f(t)\,dt NEWLINE\]NEWLINE in \(L^p(0, 1)\) is not supercyclic [\textit{E. A. Gallardo-Gutiérrez} and \textit{A. Montes-Rodríguez}, Integral Equations Oper. Theory 50, No.~2, 211--216 (2004; Zbl 1080.47011)]. The author defines this paper as a ``quest for supercyclic or even hypercyclic operators as close as possible to the Volterra operator'', ``as close as possible'' understood as ``commuting with''. The operators called into play are truncated convolution operators in \(L^p(0, 1)\) (roughly speaking, restrictions to \([0, 1]\) of convolutions with Borel measures) and the main result is that the semigroup generated by finitely many truncated co…
reviewed by: Hector O. Fattorini
description: scientific article; zbMATH DE number 6031021