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Hypercyclic $ C_0$-semigroups and evolution families generated by first order differential operators

Author: T. Kalmes
Journal: Proc. Amer. Math. Soc. 137 (2009), 3833-3848
MSC (2000): Primary 47A16, 47D06
Published electronically: June 18, 2009
MathSciNet review: 2529893
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Abstract: We show that $ C_0$-semigroups generated by first order partial differential operators on $ L^p(\Omega,\mu)$ and $ C_{0,\rho}(\Omega)$, respectively, are hypercyclic if and only if they are weakly mixing, where $ \Omega\subset\mathbb{R}^d$ is open. In the case of $ d=1$ we give an easy to check characterization of when this happens. Furthermore, we give an example of a hypercyclic evolution family such that not all of the operators of the family are hypercyclic themselves. This stands in complete contrast to hypercyclic $ C_0$-semigroups.

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T. Kalmes
Affiliation: Bergische Universität Wuppertal, FB Mathematik und Naturwissenschaften, D-42097 Wuppertal, Germany

Received by editor(s): January 23, 2009
Received by editor(s) in revised form: March 3, 2009
Published electronically: June 18, 2009
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.