Coarse topological transitivity on open cones and coarsely J-class and D-class operators |
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Authors: | Antonios Manoussos |
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Institution: | Fakultät für Mathematik, SFB 701, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany |
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Abstract: | We generalize the concept of coarse hypercyclicity, introduced by Feldman in 13], to that of coarse topological transitivity on open cones. We show that a bounded linear operator acting on an infinite dimensional Banach space with a coarsely dense orbit on an open cone is hypercyclic and a coarsely topologically transitive (mixing) operator on an open cone is topologically transitive (mixing resp.). We also “localize” these concepts by introducing two new classes of operators called coarsely J-class and coarsely D -class operators and we establish some results that may make these classes of operators potentially interesting for further studying. Namely, we show that if a backward unilateral weighted shift on l2(N) is coarsely J-class (or D -class) on an open cone then it is hypercyclic. Then we give an example of a bilateral weighted shift on l∞(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∞(Z) does not support J-class bilateral weighted shifts, see 10]. Finally, we show that there exists a non-separable Banach space which supports no coarsely D-class operators on open cones. Some open problems are added. |
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Keywords: | Coarse topological transitivity Coarse hypercyclicity Topological transitivity Hypercyclicity Coarsely J-class operator Coarsely D-class operator J-class operator Open cone |
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