The Hypercyclicity Criterion for sequences of operators
Volume 157 / 2003
Abstract
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 $T$.
