Let $S$ be the (say, left) shift operator on $\ell^2(\mathbb{Z})$. For a non-zero vector $x \in \ell^2(\mathbb{Z})$, consider the set
$$X = \{ S^n v \mid n \in \mathbb{Z} \}.$$
Is this always a total set, i.e., is its span dense in $\ell^2(\mathbb{Z})$?



Do powers of the shift operator applied to a non-zero vector always yield a total set?
Such sets are not always total. The shift operator $S$ is unitarily equivalent to multiplication by $z$ on $L^2(S^1)$. From this perspective you can see vectors for which the set you write is not total, for example the characteristic function of an interval.


