Dense range perturbations of hypercyclic operators (Q2773350)

A continuous linear operator \(T: X\to X\) defined on a Hausdorff locally convex space \(X\) is called hypercyclic if there is a vector \(x\in X\) such that the orbit \(O(T, x):= \{x,Tx,T^2 x,\dots\}\) is dense in \(X\). The author presents conditions on an operator \(S\) on \(X\) which commutes with \(T\) to ensure that the range of the operator \(T+S\) is dense in \(X\). The case of sequences of operators \((T_n)_n\) and perturbations of the form \((T_n+ S_n)_n\) is also discussed.