Issue Math. Model. Nat. Phenom. Volume 3, Number 7, 2008 Special issue dedicated to Glenn Webb Page(s) 148 - 160 DOI 10.1051/mmnp:2008047 Published online 23 October 2008

Math. Model. Nat. Phenom. Vol. 3, No. 7, 2008, pp. 148-160
DOI: 10.1051/mmnp:2008047

Hypercyclicity of Semigroups is a Very Unstable Property

W. Desch and W. Schappacher

Karl-Franzens-Universität Graz, Institut für Mathematik und wissenschaftliches Rechnen Heinrichstraße 36, A-8010 Graz, Austria

georg.desch@uni-graz.at

Published online: 23 October 2008

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

Mathematics Subject Classification. 47A16, 47D03

Key words: hypercyclic semigroups -- perturbation

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