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On chaotic $C_0$-semigroups and infinitely regular hypercyclic vectors

Author: T. Kalmes
Journal: Proc. Amer. Math. Soc. 134 (2006), 2997-3002
MSC (2000): Primary 47A16, 47D03
Published electronically: May 5, 2006
MathSciNet review: 2231625
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Abstract: A $C_0$-semigroup $\mathcal {T}=(T(t))_{t\geq 0}$ on a Banach space $X$ is called hypercyclic if there exists an element $x\in X$ such that $\{T(t)x; t\geq 0\}$ is dense in $X$. $\mathcal {T}$ is called chaotic if $\mathcal {T}$ is hypercyclic and the set of its periodic vectors is dense in $X$ as well. We show that a spectral condition introduced by Desch, Schappacher and Webb requiring many eigenvectors of the generator which depend analytically on the eigenvalues not only implies the chaoticity of the semigroup but the chaoticity of every $T(t), t>0$. Furthermore, we show that semigroups whose generators have compact resolvent are never chaotic. In a second part we prove the existence of hypercyclic vectors in $D(A^\infty )$ for a hypercyclic semigroup $\mathcal {T}$, where $A$ is its generator.

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Additional Information

T. Kalmes
Affiliation: FB IV - Mathematik, Universität Trier, D - 54286 Trier, Germany
MR Author ID: 717771

Received by editor(s): May 4, 2005
Published electronically: May 5, 2006
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.