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TRANSITIVE AND HYPERCYCLIC OPERATORS ON LOCALLY CONVEX SPACES

Published online by Cambridge University Press:  10 March 2005

J. BONET ,
L. FRERICK ,
A. PERIS  and
J. WENGENROTH
J. BONET
Affiliation:
E.T.S. Arquitectura, Departament de Matemàtica, Aplicada, Universitat Politècnica de València, E-46022 València, Spain jbonet@mat.upv.es, aperis@mat.upv.es
L. FRERICK
Affiliation:
FB Mathematik, Bergische Universität Wuppertal, Gauß-Straße 20, D-42097 Wuppertal, Germany frerick@math.uni-wuppertal.de
A. PERIS
Affiliation:
E.T.S. Arquitectura, Departament de Matemàtica, Aplicada, Universitat Politècnica de València, E-46022 València, Spain jbonet@mat.upv.es, aperis@mat.upv.es
J. WENGENROTH
Affiliation:
FB IV Mathematik, Universität Trier, D-54286 Trier, Germany wengen@uni-trier.de
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Abstract

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Fréchet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space $\varphi $ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on $\varphi$). (2) The space of all test functions for distributions, which is also a complete direct sum of Fréchet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Fréchet space contains a dense hyperplane that admits no transitive operator.

Keywords

47A16 (primary)46A0346A0446A1337D45 (secondary)

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Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 37 , Issue 2 , March 2005 , pp. 254 - 264
Copyright
© The London Mathematical Society 2005

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Footnotes

This paper was completed during a visit by L. Frerick to the Universitat Politècnica de València in late 2002 and early 2003. The support of the Programa de Incentivo a la Investigación Científica de la Universitat Politècnica de València 2002 is gratefully acknowledged. The research of J. Bonet and A. Peris was partially supported by MCYT and FEDER Proyecto no. BFM2001-2670, and by AVCIT Grup 03/050.