Two problems on weighted shifts in linear dynamics

TL;DR

This paper explores the stability and chaotic behavior of weighted shift operators in linear dynamics, establishing conditions for structural stability and identifying spaces supporting hypercyclic but not chaotic shifts.

Contribution

It characterizes when invertible bilateral weighted shifts are strongly structurally stable and identifies a sequence space with hypercyclic but non-chaotic weighted shifts.

Findings

Invertible bilateral weighted shifts are strongly structurally stable iff they have the shadowing property.

A K{"o}the sequence space supports a frequently hypercyclic weighted shift but no chaotic weighted shifts.

The paper provides new insights into the stability and chaos in weighted shift operators.

Abstract

We show that an invertible bilateral weighted shift is strongly structurally stable if and only if it has the shadowing property. We also exhibit a K{\"o}the sequence space supporting a frequently hypercyclic weighted shift, but no chaotic weighted shifts.