Let us consider $\ell^p(\mathbb{Z})$. We know that the vector $e_1=(\dots,0,0,1,0,0,\dots)$ is a cyclic vector in sense that given the right shift operator $S:(\dots,x_0,x_1,x_2,\dots)\mapsto (\dots,x_{-1},x_0,x_1,\dots)$, $$\overline{\text{span}\{S^ne_1\}_{n\in\mathbb{Z}}}=\ell^2(\mathbb{Z}).$$

I wonder if the requirement for $n\in\mathbb{Z}$ is essential. I assume that we can find a vector $\vec{c}\in\ell^2(\mathbb{Z})$ such that

$$\overline{\text{span}\{S^n\vec c\}_{n\in\mathbb{N}}}=\ell^2(\mathbb{Z}).$$

I think that using $H^2(\mathbb{T})$ can be useful but I don't know how to use it. Am I right? Can you give an example of such vector?