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2 results

  • Hypercyclic and mixing operator semigroups

  • Part of
  • Stanislav Shkarin
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 54 / Issue 3 / October 2011
    Published online by Cambridge University Press:
    27 June 2011, pp. 761-782
    • Article
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  • We describe a class of topological vector spaces admitting a mixing uniformly continuous operator group with holomorphic dependence on the parameter t. This result builds on those existing in the literature. We also describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups .

  • Hypercyclicity of Semigroups is a Very Unstable Property

  • W. Desch, W. Schappacher
  • Journal:
    Mathematical Modelling of Natural Phenomena / Volume 3 / Issue 7 / 2008
    Published online by Cambridge University Press:
    23 October 2008, pp. 148-160
    Print publication:
    2008

  • 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.