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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