Existence and nonexistence of hypercyclic semigroups

Bernal-González, L.; Grosse-Erdmann, K.-G.

doi:10.1090/S0002-9939-06-08524-8

Existence and nonexistence of hypercyclic semigroups
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by L. Bernal-González and K.-G. Grosse-Erdmann PDF

Proc. Amer. Math. Soc. 135 (2007), 755-766 Request permission

Abstract:

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from—and considerably shorter than—the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Fréchet space. This complements recent results due to Bès and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.

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