Hypercyclic 𝐶₀-semigroups and evolution families generated by first order differential operators

Kalmes, T.

doi:10.1090/S0002-9939-09-09955-9

Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators
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Proc. Amer. Math. Soc. 137 (2009), 3833-3848 Request permission

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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