Topologically mixing hypercyclic operators. (English) Zbl 1054.47006

Costakis, George; Sambarino, Martín

Proc. Am. Math. Soc. 132, No. 2, 385-389 (2004).

Authors’ abstract: “Let \(X\) be a separable Fréchet space. We prove that a linear operator \(T: X\to X\) satisfying a special case of the hypercyclicity criterion is topologically mixing, i.e., for any given open sets \(U\), \(V\) there exists a positive integer \(N\) such that \(T^n(U)\cap V\neq\emptyset\) for any \(n\geq N\). We also characterize those weighted backward shift operators that are topologically mixing.”

Reviewer: Gerd Herzog (Karlsruhe)

MSC:

47A16	Cyclic vectors, hypercyclic and chaotic operators
47B37	Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
37A25	Ergodicity, mixing, rates of mixing

Keywords:

hypercyclic operators; hypercyclicity criterion; topologically mixing