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 $\{ ({ \bf 1}+a)^na:n\in\N\}$ are both dense in $A$, where $\bf 1$ 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 $\sigma(T)=\{ 1\}$.
Categories: math.FA math.DS