Chaotic Banach algebras

Abstract

We construct an infinite dimensional non-unital Banach algebra $A$ and $a\in A$ such that the sets $\{za^n:z\in\C,\ n\in\N\}$ and $\{(1+a{)}^{n}a:n\in \mathbb{N}\}$ are both dense in $A$, where $1$ is the unity in the unitalization $A^{\#}=A\oplus \spann\{{\bf 1}\}$ of $A$. As a byproduct, we get a hypercyclic operator $T$ on a Banach space such that $T\oplus T$ is non-cyclic and $\sigma (T)=\{1\}$.

Under consideration in Journal of Functional Analysis