Epsilon-hypercyclic operators

0374204 - MÚ 2012 RIV US eng J - Journal Article
Badea, C. - Grivaux, S. - Müller, Vladimír
Epsilon-hypercyclic operators.
Ergodic Theory and Dynamical Systems. Roč. 30, č. 6 (2010), s. 1597-1606. ISSN 0143-3857. E-ISSN 1469-4417
R&D Projects: GA ČR(CZ) GA201/06/0128
Institutional research plan: CEZ:AV0Z10190503
Keywords : orbits * dense
Subject RIV: BA - General Mathematics
Impact factor: 0.795, year: 2010
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7921209

Let X be a separable infinite-dimensional Banach space, and T a bounded linear operator on X; T is hypercyclic if there is a vector x in X with dense orbit under the action of T. For a fixed epsilon is an element of (0, 1), we say that T is epsilon-hypercyclic if there exists a vector x in X such that for every non-zero vector y is an element of X there exists an integer n with parallel to T(n)x - y parallel to <= epsilon parallel to y parallel to. The main result of this paper is a construction of a bounded linear operator T on the Banach space l(1) which is epsilon-hypercyclic without being hypercyclic. This answers a question from V. Muller [Three problems, Mini-Workshop: Hypercyclicity and linear chaos, organized by T. Bermudez, G. Godefroy, K.-G. Grosse-Erdmann and A. Peris. Oberwolfach Rep. 3 (2006), 2227-2276].
Permanent Link: http://hdl.handle.net/11104/0207176