Perturbations of hypercyclic vectors. (Q1856818)

A bounded linear operator \(T\) on a separable Banach space \(\mathcal B\) is said to be hypercyclic if there is \(x \in \mathcal B\), also called hypercyclic, such that the elements in the orbit \(\{T^ n x\}_{n\geq 0}\) are dense in \(\mathcal B\). Hypercyclicity is one of the strongest forms of cyclicity. \textit{S. Rolewicz} [Stud. Math. 32, 17--22 (1969; Zbl 0174.44203)] proved that hypercyclic operators cannot exist in finite dimensional spaces. The author of the present paper proves that for fixed \(d>0\), if the orbit of a vector \(x\) under \(T\) falls into the ball of center \(y\) and radius \(d\) for each \(y \in \mathcal B\), then the operator \(T\) is hypercyclic while \(x\) itself may not. The proof of this nice result is mainly based on the non-compactness of the unit ball of an infinite-dimensional Banach space. The author also proves that perturbations \(x+y\) of a hypercyclic vector \(x\), where \(y\) has an orbit which is relatively compact, are also hypercyclic.