J-class operators and hypercyclicity

Details J-class operators and hypercyclicity J-class operators and hypercyclicity

2007-04-25

arxiv.org/abs/0704.3354v2

Mathematics

Functional Analysis

21 pages. Shorter version submitted for publication

Scientific paper

The purpose of the present work is to treat a new notion related to linear dynamics, which can be viewed as a "localization" of the notion of hypercyclicity. In particular, let $T$ be a bounded linear operator acting on a Banach space $X$ and let $x$ be a non-zero vector in $X$ such that for every open neighborhood $U\subset X$ of $x$ and every non-empty open set $V\subset X$ there exists a positive integer $n$ such that $T^{n}U\cap V\neq\emptyset$. In this case $T$ will be called a $J$-class operator. We investigate the class of operators satisfying the above property and provide various examples. It is worthwhile to mention that many results from the theory of hypercyclic operators have their analogues in this setting. For example we establish results related to the Bourdon-Feldman theorem and we characterize the $J$-class weighted shifts. We would also like to stress that even non-separable Banach spaces which do not support topologically transitive operators, as for example $l^{\infty}(\mathbb{N})$, do admit $J$-class operators.

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