Frequently Hypercyclic Bilateral Shifts

Abstract

It is not known if the inverse of a frequently hypercyclic bilateral weighted shift on c 0 (Z) is again frequently hypercyclic. We show that the corresponding problem for upper frequent hypercyclicity has a positive answer. We characterise, more generally, when bilateral weighted shifts on Banach sequence spaces are (upper) frequently hypercyclic.

Zero-one law of orbital limit points for weighted shifts

“…We do not know whether, in general, the conclusion in Theorem 3.1 implies that B w is F-hypercyclic. The conclusion falls well short of the known characterization of Fhypercyclicity, see [11,Theorem 15].…”

Linear Dynamical Systems

“…With the general machinery of A-hypercyclicity, the results in [41], [54] and [68] for U-frequent and reiterative hypercyclicity follow as special cases. This approach was also employed by Grosse-Erdmann [97] to prove the following theorem. Theorem 3.12 (Grosse-Erdmann [97]).…”

“…This approach was also employed by Grosse-Erdmann [97] to prove the following theorem. Theorem 3.12 (Grosse-Erdmann [97]). Let B w be an invertible weighted shift on c 0 (Z).…”

Inverse of U-frequently hypercyclic operators

“…The purpose of this paper is to answer the following open question which can be found in [3,5,11,12]:…”

“…The purpose of this paper is to answer the following open question which can be found in [3,5,11,12]: Does the inverse of an invertible frequently hypercyclic operator is frequently hypercyclic? It is well-known that the inverse of an invertible hypercyclic operator is always hypercyclic.…”