The Hypercyclicity Criterion for sequences of operators

L. Bernal-González, and K.-G. Grosse-Erdmann. "The Hypercyclicity Criterion for sequences of operators." Studia Mathematica 157.1 (2003): 17-32. <http://eudml.org/doc/284569>.

@article{L2003,
abstract = {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 $(T_\{n_\{k\}\})$, 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.},
author = {L. Bernal-González, K.-G. Grosse-Erdmann},
journal = {Studia Mathematica},
keywords = {hypercyclic operator; hypercyclicity criterion; chaotic operator},
language = {eng},
number = {1},
pages = {17-32},
title = {The Hypercyclicity Criterion for sequences of operators},
url = {http://eudml.org/doc/284569},
volume = {157},
year = {2003},
}

TY - JOUR
AU - L. Bernal-González
AU - K.-G. Grosse-Erdmann
TI - The Hypercyclicity Criterion for sequences of operators
JO - Studia Mathematica
PY - 2003
VL - 157
IS - 1
SP - 17
EP - 32
AB - 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 $(T_{n_{k}})$, 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.
LA - eng
KW - hypercyclic operator; hypercyclicity criterion; chaotic operator
UR - http://eudml.org/doc/284569
ER -