Epsilon-hypercyclic operators

Abstract

AbstractLet 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 ε∈(0,1), we say that T is ε-hypercyclic if there exists a vector x in X such that for every non-zero vector y∈X there exists an integer n with $\|T^nx-y\|\leq \varepsilon \|y\|$. The main result of this paper is a construction of a bounded linear operator T on the Banach space ℓ1 which is ε-hypercyclic without being hypercyclic. This answers a question from V. Müller [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].