Existence of hypercyclic operators on topological vector spaces (Q1368851)

The author establishes the following result: Suppose \(X\) is a topological vector space with a bounded biorthogonal system \(\{x_n, f_n\}\). Suppose \(X\) is \(\ell^1\)-complete with respect to \((x_n)\). (a) If \(\widetilde T: X\to X\) is continuous and \(\widetilde T(x_n)= w_nx_{n-1}\) for some bounded sequence \((w_n)\) of positive real numbers, then \(I_X+\widetilde T\) is hypercyclic. (b) If \(X\) is locally convex and the dual space of \(X\) is Mackey complete, then \(X\) admits a hypercyclic operator. (c) If \(X\) is locally convex with respect to \((x_n)\) and \((f_n)\) is equicontinuous, then \(X\) admits a hypercyclic operator. It follows from this result that every Banach space admits a bounded hypercyclic operator. This answers a question of \textit{S. Rolewicz} in the affirmative [Stud. Math. 32, 17-22 (1969; Zbl 0174.44203)].