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Journal of Operator Theory

Volume 62, Issue 2, Fall 2009  pp. 341-355.

The hypercyclicity criterion and hypercyclic sequences of multiples of operators

Authors George Costakis (1) and Demetris Hadjiloucas (2)
Author institution: (1) Department of Mathematics, University of Crete, Knossos Avenue, GR-714 09, Heraklion, Crete, Greece
(2) Department of Computer Science and Engineering, European University Cyprus, 6 Diogenes Street, Engomi, P.O.Box 22006, 1516 Nicosia, Cyprus

Summary:  Let $T$ be a linear continuous operator acting on a Banach space $X$ and $\{\lambda_n\}$ a sequence of non-zero complex numbers satisfying $\frac{\lambda_{n+1}}{\lambda_n}\to 1$. In this article we look at sequences of operators of the form $\{\lambda_n T^n\}$. In earlier work we showed that under the assumption that $T$ is hypercyclic, if for some $x\in X$ the set $\{\lambda_n T^n x: n\in\mathbb{N}\}$ is somewhere dense then it is everywhere dense, a Bourdon--Feldman type theorem. In this article we show that this result fails to hold if the assumption of hypercyclicity for $T$ is removed. A condition for the sequence $\{\lambda_n\}$ under which an Ansari type theorem holds, namely, if $\{\lambda_n T^n\}$ is hypercyclic then $\{\lambda_n T^{kn}\}$ is hypercyclic for $k=2,3,\ldots$, is given. We show that if this condition is not satisfied, the result may fail to hold. Furthermore, we establish equivalences to the hypercyclicity criterion for this class of operator sequences.

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