A weighted bilateral shift with cyclic square is supercyclic

TL;DR

This paper investigates the properties of weighted bilateral shift operators on bclp(a) spaces, establishing equivalences among supercyclicity and cyclicity, and providing new conditions for cyclicity, especially for compact shifts.

Contribution

It introduces a new sufficient condition for cyclicity of weighted bilateral shifts and shows the equivalence of supercyclicity and cyclicity properties under certain conditions.

Findings

Supercyclicity, weak supercyclicity, and cyclicity are equivalent for these operators.

A new sufficient condition for cyclicity of weighted bilateral shifts is established.

Any compact weighted bilateral shift is cyclic.

Abstract

It is shown that for a bounded weighted bilateral shift $T$ acting on $\ell_p(\Z)$ for $1\leq p\leq 2$ supercyclicity of $T$, weak supercyclicity of $T$, cyclicity of $T\oplus T$ and cyclicity of $T^2$ are equivalent. A new sufficient condition for cyclicity of a weighted bilateral shift is proved, which implies, in particular, that any compact weighted bilateral shift is cyclic.