Transitive and Hypercyclic Operators on Locally Convex Spaces
E.T.S. Arquitectura, Departament de Matemàtica, Aplicada, Universitat Politècnica de València E-46022 València, Spain jbonet{at}mat.upv.es, aperis{at}mat.upv.es
FB Mathematik, Bergische Universität Wuppertal Gauß-Straße 20, D-42097 Wuppertal, Germany frerick{at}math.uni-wuppertal.de
FB IV Mathematik, Universität Trier D-54286 Trier, Germany wengen{at}uni-trier.de
Received 18 July 2003. Revision received 2 February 2004.
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 
of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on . (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. 2000 Mathematics Subject Classification 47A16 (primary), 46A03, 46A04, 46A13, 37D45 (secondary).
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.