Journal of Operator Theory

Volume 52, Issue 1, Summer 2004 pp. 39-59.

A weakly hypercyclic operator that is not norm hypercyclic

Authors: Kit C. Chan (1) and Rebecca Sanders (2)
Author institution: (1) Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403 USA
(2) Department of Mathematics, Statistics and Computer Sciences, Marquette University, Milwaukee, WS 53201 USA

Summary: We give a sufficient condition for a bilateral weighted shift to be hypercyclic in the weak topology. Using this condition, we provide one such shift that fails to be hypercyclic in the norm topology. Even more interesting, the shift is bounded below by 1 and consequently every vector has a norm increasing orbit. This result provides a negative answer to a natural question raised by Feldman who asked whether every weakly hypercyclic operator is necessarily norm hypercyclic. On the other hand, if the operator is a unilateral weighted backward shift, we prove the answer is positive. Furthermore, with a simple condition on the weights, there exists a weakly hypercyclic vector that is not a norm hypercyclic vector.

Keywords: hypercyclic operator, hypercyclic vector, bilateral weighted shift, unilateral weighted backward shift, weak topology

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