We show that under no hypotheses on the density of the ranges of the mappings involved, an almost-commuting sequence (Tₙ) of operators on an F-space X satisfies the Hypercyclicity Criterion if and only if it has a hereditarily hypercyclic subsequence , and if and only if the sequence (Tₙ ⊕ Tₙ) is hypercyclic on X × 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 (Tₙ) 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 (Tⁿ) of a single operator T.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-1-2,
author = {L. Bernal-Gonz\'alez and K.-G. Grosse-Erdmann},
title = {The Hypercyclicity Criterion for sequences of operators},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {17-32},
zbl = {1032.47006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-1-2}
}
L. Bernal-González; K.-G. Grosse-Erdmann. The Hypercyclicity Criterion for sequences of operators. Studia Mathematica, Tome 157 (2003) pp. 17-32. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-1-2/