The Hypercyclicity Criterion for sequences of operators

Tom 157 / 2003

L. Bernal-González, K.-G. Grosse-Erdmann Studia Mathematica 157 (2003), 17-32 MSC: Primary 47A16; Secondary 46A16, 47B33, 54H20. DOI: 10.4064/sm157-1-2


We show that under no hypotheses on the density of the ranges of the mappings involved, an almost-commuting sequence $(T_n)$ of operators on an F-space $X$ satisfies the Hypercyclicity Criterion if and only if it has a hereditarily hypercyclic subsequence $(T_{n_k})$, and if and only if the sequence $(T_n \oplus T_n)$ is hypercyclic on $X \times X$. This strengthens and extends a recent result due to Bès and Peris. We also find a new characterization of the Hypercyclicity Criterion in terms of a condition introduced by Godefroy and Shapiro. Finally, we show that a weakly commuting hypercyclic sequence $(T_n)$ satisfies the Hypercyclicity Criterion whenever it has a dense set of points with precompact orbits. We remark that some of our results are new even in the case of iterates $(T^n)$ of a single operator $T$.


  • L. Bernal-GonzálezDepartamento de Análisis Matemático
    Facultad de Matemáticas, Apdo. 1160
    Universidad de Sevilla
    Avda. Reina Mercedes
    41080 Sevilla, Spain
  • K.-G. Grosse-ErdmannFachbereich Mathematik
    Fern Universität Hagen
    D-58084 Hagen, Germany

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