# Coarse topological transitivity on open cones and coarsely J-class and D-class operators

Manoussos A (2014)
Journal of Mathematical Analysis and Applications 413(2): 715-726.

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Journal Article | Published | English

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We generalize the concept of coarse hypercyclicity, introduced by Feldman in\cite{Fe1}, to that of coarse topological transitivity on open cones. We showthat a bounded linear operator acting on an infinite dimensional Banach spacewith a coarsely dense orbit on an open cone is hypercyclic and a coarselytopologically transitive (mixing) operator on an open cone is topologicallytransitive (mixing resp.). We also "localize" these concepts by introducing twonew classes of operators called coarsely $J$-class and coarsely $D$-classoperators and we establish some results that may make these classes ofoperators potentially interesting for further studying. Namely, we show that ifa backward unilateral weighted shift on $l^2(\mathbb{N})$ is coarsely $J$-class(or $D$-class) on an open cone then it is hypercyclic. Then we give an exampleof a bilateral weighted shift on $l^{\infty}(\mathbb{Z})$ which is coarsely$J$-class, hence it is coarsely $D$-class, and not $J$-class. Note that,concerning the previous result, it is well known that the space$l^{\infty}(\mathbb{Z})$ does not support $J$-class operators, see\cite{CosMa2}. Finally, we show that there exists a non-separable Banach spacewhich supports no coarsely $D$-class operators on open cones. Some openproblems are added.
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Manoussos A. Coarse topological transitivity on open cones and coarsely J-class and D-class operators. Journal of Mathematical Analysis and Applications. 2014;413(2):715-726.
Manoussos, A. (2014). Coarse topological transitivity on open cones and coarsely J-class and D-class operators. Journal of Mathematical Analysis and Applications, 413(2), 715-726. doi:10.1016/j.jmaa.2013.12.038
Manoussos, A. (2014). Coarse topological transitivity on open cones and coarsely J-class and D-class operators. Journal of Mathematical Analysis and Applications 413, 715-726.
Manoussos, A., 2014. Coarse topological transitivity on open cones and coarsely J-class and D-class operators. Journal of Mathematical Analysis and Applications, 413(2), p 715-726.
A. Manoussos, “Coarse topological transitivity on open cones and coarsely J-class and D-class operators”, Journal of Mathematical Analysis and Applications, vol. 413, 2014, pp. 715-726.
Manoussos, A.: Coarse topological transitivity on open cones and coarsely J-class and D-class operators. Journal of Mathematical Analysis and Applications. 413, 715-726 (2014).
Manoussos, Antonios. “Coarse topological transitivity on open cones and coarsely J-class and D-class operators”. Journal of Mathematical Analysis and Applications 413.2 (2014): 715-726.
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arXiv 1306.5331