Mathematische Zeitschrift

, Volume 236, Issue 4, pp 779–786

Multi-hypercyclic operators are hypercyclic

  • Alfredo Peris
Orignial article

DOI: 10.1007/PL00004850

Cite this article as:
Peris, A. Math Z (2001) 236: 779. doi:10.1007/PL00004850

Abstract.

Herrero conjectured in 1991 that every multi-hypercyclic (respectively, multi-supercyclic) operator on a Hilbert space is in fact hypercyclic (respectively, supercyclic). In this article we settle this conjecture in the affirmative even for continuous linear operators defined on arbitrary locally convex spaces. More precisely, we show that, if \(T:E \rightarrow E\) is a continuous linear operator on a locally convex space E such that there is a finite collection of orbits of T satisfying that each element in E can be arbitrarily approximated by a vector of one of these orbits, then there is a single orbit dense in E. We also prove the corresponding result for a weaker notion of approximation, called supercyclicity .

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Alfredo Peris
    • 1
  1. 1.E.T.S. Arquitectura, Universidad Politécnica de Valencia, Departamento de Matemática Aplicada, E-46071 Valencia, Spain (e-mail: aperis@pleiades.upv.es) ES