Journal of Operator Theory
Volume 72, Issue 1, Summer 2014 pp. 15-40.
Weighted shifts and disjoint hypercyclicity
Authors:
Juan Bes (1), Ozgur Martin (2),
and Rebecca Sanders (3)
Author institution: (1) Department of Mathematics and Statistics,
Bowling Green State University, Bowling Green, 43403, U.S.A.
(2) Mathematics Department, Mimar Sinan Fine Arts
University, Silahsor Cad. 71, Bomonti Sisli 34380, Istanbul,
Turkey
(3) Department of Mathematics, Statistics, and Computer Science, Marquette
University, Milwaukee, 53201, U.S.A.
Summary: 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.
DOI: http://dx.doi.org/10.7900/jot.2012aug20.1970
Keywords: hypercyclic vectors, hypercyclic operators, unilateral
weighted backward shift, bilateral weighted shift
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