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 .
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received October 18, 1999 / Published online February 5, 2001
Rights and permissions
About this article
Cite this article
Peris, A. Multi-hypercyclic operators are hypercyclic. Math Z 236, 779–786 (2001). https://doi.org/10.1007/PL00004850
Issue Date:
DOI: https://doi.org/10.1007/PL00004850