Mathematics and Statistics Faculty Publications

Weighted Shifts and Disjoint Hypercyclicity

Document Type

Article

Abstract

We give characterizations for finite collections of disjoint hypercyclic weighted shift operators, both in the unilateral and bilateral cases. It follows that some well-known results about the dynamics of an operator fail to hold true in the disjoint setting. For example, finite collections of disjoint hypercyclic shifts never satisfy the disjoint hypercyclicity criterion, even though they satisfy the disjoint blow-up/collapse property; thus they are densely disjoint hypercyclic, but are never hereditarily densely disjoint hypercyclic. Moreover, they fail to be disjoint weakly mixing. Also, any finite collection of bilateral shifts containing an invertible shift fails to be disjoint hypercyclic. Even more, each of these facts is in sharp contrast with what happens to finite collections of shift operators raised to positive, distinct powers.

Publication Date

Summer 2014

Publication Title

Journal of Operator Theory

Publisher

Theta Foundation

DOI

https://doi.org/10.7900/jot.2012aug20.1970

Volume

72

Issue

1

Start Page No.

15

End Page No.

40

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