Journal of Operator Theory

Volume 54, Issue 1, Fall 2005 pp. 147-168.

Hypercyclic operators, mixing operators, and the Bounded Steps Problem

Authors: Sophie Grivaux
Author institution:Laboratoire Paul Painlevé, UMR 8524, Université des Sciences et Technologies de Lille, Cité Scientifique, 59655 Villeneuve d'Ascq Cedex, France

Summary: The main topic of this paper is the Hypercyclicity Criterion. We construct mixing operators which fail Kitai's Criterion, thus answering a question of Shapiro. We show that the Bounded Steps Problem introduced by Bès and Peris is equivalent to the Hypercyclicity Criterion. Then we show that hypercyclic operators which satisfy an additional regularity assumption satisfy the Hypercyclicity Criterion: if $T$ is hypercyclic and upper-triangular, or if $T$ has a dense set of vectors with bounded orbit, or if $T \oplus T$ is cyclic, then $T\oplus T$ is hypercyclic. We give similar results for cyclic operators.

Keywords: Hypercyclic operators, Hyperciclicity Creterion, mixing operators, Kitai's Criterion, direct sums of cyclic and hypercyclic operators.

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