Math. Model. Nat. Phenom.
Volume 3, Number 7, 2008Special issue dedicated to Glenn Webb
|Page(s)||148 - 160|
|Published online||23 October 2008|
Hypercyclicity of Semigroups is a Very Unstable Property
Karl-Franzens-Universität Graz, Institut für Mathematik und wissenschaftliches Rechnen
Heinrichstraße 36, A-8010 Graz, Austria
Corresponding author: email@example.com
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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