Mathematical Modelling of Natural Phenomena

Table of Contents - 2008 - Volume 3, Issue 07 (Special issue dedicated to Glenn Webb)  

Research Article

Hypercyclicity of Semigroups is a Very Unstable Property

W. Descha1 and W. Schappachera1

a1 Karl-Franzens-Universität Graz, Institut für Mathematik und wissenschaftliches Rechnen Heinrichstraße 36, A-8010 Graz, Austria

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.

(Online publication October 23 2008)

Key Words:

  • hypercyclic semigroups;
  • perturbation

Mathematics Subject Classification:

  • 47A16;
  • 47D03

Correspondence:

c1 georg.desch@uni-graz.at

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