J-class operators and hypercyclicity

Abstract

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 $T^{n}U\cap V\ne \varnothing$. 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.

21 pages. Shorter version submitted for publication

Cited by in corpus (10)

• J-class sequences of linear operators
• The Spectrum of Operators on C(K) with the Grothendieck Property and Characterization of J-Class Operators which are adjoints
• Limit and extended limit sets of matrices in Jordan normal form
• J-Class Abelian Semigroups of Matrices on C^n and Hypercyclicity
• J-class weighted shifts on the space of bounded sequences of complex numbers
• Coarse topological transitivity on open cones and coarsely J-class and D-class operators
• Dynamics of perturbations of the identity operator by multiples of the backward shift on $l^{\infty }(\mathbb{N})$
• On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple
• Limits of J-class operators
• On the existence of J-class operators