Geodesic
Hypercyclicity of Semigroups is a Very Unstable Property
W. Desch1 ; W. Schappacher1
1 Karl-Franzens-Universität Graz, Institut für Mathematik und wissenschaftliches Rechnen Heinrichstraße 36, A-8010 Graz, Austria
Mathematical modelling of natural phenomena, Tome 3 (2008) no. 7, pp. 148-160

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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.
DOI : 10.1051/mmnp:2008047

W. Desch 1 ; W. Schappacher 1

1 Karl-Franzens-Universität Graz, Institut für Mathematik und wissenschaftliches Rechnen Heinrichstraße 36, A-8010 Graz, Austria 
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W. Desch; W. Schappacher. Hypercyclicity of Semigroups is a Very Unstable Property. Mathematical modelling of natural phenomena, Tome 3 (2008) no. 7, pp. 148-160. doi: 10.1051/mmnp:2008047

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