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 Painlev\'e, UMR 8524, Universit\'e des Sciences et Technologies de Lille, Cit\' e 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 B\` es 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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