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The Hypercyclicity Criterion for sequences of operators

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Bernal-González L., Grosse-Erdmann K.-G.

Studia Mathematica

2003

tom 157

nr 1

17-32

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.

Ograniczanie wyników

1 Studia Mathematica

1 Bernal-González L.

1 Grosse-Erdmann K.-G.

1 2003

Wyniki wyszukiwania

The Hypercyclicity Criterion for sequences of operators