Integral Equations and Operator Theory

, Volume 51, Issue 2, pp 275–281

Syndetically Hypercyclic Operators

Original Paper

DOI: 10.1007/s00020-003-1253-9

Cite this article as:
Peris, A. & Saldivia, L. Integr. equ. oper. theory (2005) 51: 275. doi:10.1007/s00020-003-1253-9

Abstract.

Given a continuous linear operator T ∈ L(x) defined on a separable \(\mathcal{F}\) -space X, we will show that T satisfies the Hypercyclicity Criterion if and only if for any strictly increasing sequence of positive integers \(\{ n_k \} _k \) such that \(\sup _k \{ n_{k + 1} - n_k \} < \infty ,\) the sequence \(\{ T^{n_k } \} _k \) is hypercyclic. In contrast we will also prove that, for any hypercyclic vector x ∈ X of T, there exists a strictly increasing sequence \(\{ n_k \} _k \) such that \(\sup _k \{ n_{k + 1} - n_k \} = 2\) and \(\{ T^{n_k } x\} _k \) is somewhere dense, but not dense in X. That is, T and \(\{ T^{n_k } \} _k \) do not share the same hypercyclic vectors.

Mathematics Subject Classification (2000).

Primary 47A16Secondary 37D4546A04

Keywords.

Hypercyclic vectorshypercyclicity criterionweakly mixing

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.E.T.S. Arquitectura, Departament de Matemàtica AplicadaUniversitat Politècnica de ValènciaValènciaSpain
  2. 2.Mathematics DepartmentMichigan State UniversityEast LansingU.S.A