Hypercyclicity of Semigroups is a Very Unstable Property

W. Desch; W. Schappacher

Mathematical Modelling of Natural Phenomena (2008)

  • Volume: 3, Issue: 7, page 148-160
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Desch, W., and Schappacher, W.. "Hypercyclicity of Semigroups is a Very Unstable Property." Mathematical Modelling of Natural Phenomena 3.7 (2008): 148-160. <http://eudml.org/doc/222341>.

@article{Desch2008,
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. },
author = {Desch, W., Schappacher, W.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {hypercyclic semigroups; perturbation},
language = {eng},
month = {10},
number = {7},
pages = {148-160},
publisher = {EDP Sciences},
title = {Hypercyclicity of Semigroups is a Very Unstable Property},
url = {http://eudml.org/doc/222341},
volume = {3},
year = {2008},
}

TY - JOUR
AU - Desch, W.
AU - Schappacher, W.
TI - Hypercyclicity of Semigroups is a Very Unstable Property
JO - Mathematical Modelling of Natural Phenomena
DA - 2008/10//
PB - EDP Sciences
VL - 3
IS - 7
SP - 148
EP - 160
AB - 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.
LA - eng
KW - hypercyclic semigroups; perturbation
UR - http://eudml.org/doc/222341
ER -

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