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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On chaotic $C_0$-semigroups and infinitely regular hypercyclic vectors
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by T. Kalmes PDF
Proc. Amer. Math. Soc. 134 (2006), 2997-3002 Request permission

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
  • Email: kalm4501@uni-trier.de
  • Received by editor(s): May 4, 2005
  • Published electronically: May 5, 2006
  • Communicated by: Jonathan M. Borwein
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 2997-3002
  • MSC (2000): Primary 47A16, 47D03
  • DOI: https://doi.org/10.1090/S0002-9939-06-08391-2
  • MathSciNet review: 2231625