Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 pp. 323-345.

Hypercyclic shift factorizations for bilateral weighted shift operators

Authors: Kit C. Chan (1), Rebecca Sanders (2)
Author institution: (1) Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, 43403, U.S.A.
(2) Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, 53201, U.S.A.

Summary: Taking the perspective that a bilateral weighted shift is an operator that shifts some two-sided canonical basic sequence of $\ell^p(\mathbb{Z})$, with $1\leqslant p < \infty$, we show that every bilateral weighted shift on $\ell^p(\mathbb{Z})$ has a factorization $T = AB$, where $A$ and $B$ are hypercyclic bilateral weighted shifts. For the case when $T$ is invertible, both shifts $A$ and $B$ may be selected to be invertible as well. Moreover, we show analogous hypercyclic factorization results for diagonal operators with nonzero diagonal entries.

DOI: http://dx.doi.org/10.7900/jot.2019jul22.2284
Keywords: weighted shift operator, hypercyclic operator, diagonal operator, factorization

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