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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:413

Article

Manoussos, Antonios

关键词: Coarse topological transitivity;    Coarse hypercyclicity;    Topological transitivity;    Hypercyclicity;    Coarsely J-class operator;    Coarsely D-class operator;    J-class operator;    Open cone;   

DOI  :  10.1016/j.jmaa.2013.12.038

来源: Elsevier

【 摘 要 】

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 l(2)(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(infinity)(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(infinity) (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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free