Journal of Operator Theory

Volume 61, Issue 1, Winter 2009 pp. 191-223.

Two criteria for a path of operators to have common hypercyclic vectors

Authors: Kit C. Chan
Author institution: Dept. of Mathematics and Statistics, Bowling Green State University, Bowling Green,43403, USA

Summary: We offer two conditions for a path of bounded linear operators on a Banach space to have a dense $G_\delta$ set of common hypercyclic vectors. One of them is an equivalent condition and the other one is a generalization of the hypercyclicity criterion. Using the conditions, we show that between any two hypercyclic unilateral weighted backward shifts, there exists a path of such operators having a dense $G_\delta$ set of common hypercyclic vectors. Furthermore, we prove that such a set of vectors exists for a path of scalar multiples of the unweighted shift, reproducing a result of Abakumov and Gordon, and of Costakis and Sambarino. Motivated by our results, we provide an example of a path of unilateral weighted backward shifts that fails to have any common hypercyclic vector. Lastly, we adopt the main results to bilateral weighted shifts.

Keywords: Hypercyclic operator, hypercyclic vector, unilateral weighted backward shift

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