Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators
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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.References
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Additional Information
- T. Kalmes
- Affiliation: Bergische Universität Wuppertal, FB Mathematik und Naturwissenschaften, D-42097 Wuppertal, Germany
- MR Author ID: 717771
- Email: kalmes@math.uni-wuppertal.de
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3833-3848
- MSC (2000): Primary 47A16, 47D06
- DOI: https://doi.org/10.1090/S0002-9939-09-09955-9
- MathSciNet review: 2529893