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Hypercyclicity of Semigroups is a Very Unstable Property

Published online by Cambridge University Press:  23 October 2008

W. Desch  and
W. Schappacher
W. Desch
Affiliation:
Karl-Franzens-Universität Graz, Institut für Mathematik und wissenschaftliches Rechnen Heinrichstraße 36, A-8010 Graz, Austria
W. Schappacher*
Affiliation:
Karl-Franzens-Universität Graz, Institut für Mathematik und wissenschaftliches Rechnen Heinrichstraße 36, A-8010 Graz, Austria
*
georg.desch@uni-graz.at
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Abstract

Hypercyclicity of C0-semigroups is a very unstable property: We give examples toshow that adding arbitrary small constants or a bounded rank one operator to the generator of ahypercyclic semigroup can destroy hypercyclicity. Also the limit of hypercyclic semigroups (evenin operator norm topology) need not be hypercyclic, and a hypercyclic semigroup can be the limitof nonhypercyclic ones. Hypercyclicity is not inherited by the Yosida approximations. Finally, therestriction of a hypercyclic nonnegative semigroup in a Banach lattice to the positive cone may befar from hypercyclic.

Keywords

hypercyclic semigroupsperturbation

Information

Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 3 , Issue 7: Special issue dedicated to Glenn Webb , 2008 , pp. 148 - 160
Copyright
© EDP Sciences, 2008

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