The Hypercyclicity Criterion for sequences of operators

L. Bernal-González ¹   ; K.-G. Grosse-Erdmann ²

¹ Departamento de Análisis Matemático Facultad de Matemáticas, Apdo. 1160 Universidad de Sevilla Avda. Reina Mercedes 41080 Sevilla, Spain
² Fachbereich Mathematik Fern Universität Hagen D-58084 Hagen, Germany

Studia Mathematica, Tome 157 (2003) no. 1, pp. 17-32

Résumé

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 $T$.

Détail

DOI : 10.4064/sm157-1-2

Keywords: under hypotheses density ranges mappings involved almost commuting sequence operators f space satisfies hypercyclicity criterion only has hereditarily hypercyclic subsequence and only sequence oplus hypercyclic times strengthens extends recent result due peris characterization hypercyclicity criterion terms condition introduced godefroy shapiro finally weakly commuting hypercyclic sequence satisfies hypercyclicity criterion whenever has dense set points precompact orbits remark results even iterates single operator