Suppose we look at $\ell^2(\mathbb{Z})$, which contains vectors $c=(\dots,c_{-2},c_{-1},c_0,c_1,\dots)$ with $\sum|c(n)|^2<\infty$.
Define the right-translation operator by $$R:\{c(n)\}\mapsto \{c(n+1)\}$$ and the left translation operator is defined similarly by $$L=R^{-1}:\{c(n)\}\mapsto \{c(n-1)\}$$.
We know that taking $$e_1=(\dots,0,0,1,0,0,0,\dots),$$ $\{R^n(e_1)\}_{n\in\mathbb{Z}}$ is a complete system.
Can one construct a vector $\vec{c}$ such that $\{R^n(\vec c)\}_{n\ge 0}$ is complete? That is, does $R$ have a cyclic vector? Does it have anything with $H^2$?