Relationships between Cyclic and Hypercyclic Operators

Abstract

A bounded linear operator $T$ on a Banach space $X$ is called hypercyclic if there exists a vector $x\in X$ such that $orb(x,T)$ is dense in $X$. The Hypercyclicity Criterion is a well-known sufficient condition for an operator to be hypercyclic. One open problem is whether there exists a space where the Hypercyclicity Criterion is also a necessary condition. For a number of reasons, the spaces with very-few operators are some natural candidates to be a positive answer to that problem. In this paper, we provide a theorem that establishes some relationships for operators in these spaces.

7 pages

References in corpus (1)

• Unconditionally saturated Banach space with the scalar-plus-compact property