Suppose $x$ is a non-zero vector in a Banach space, and $T$ is a fixed operator. Is the following true:
For any $\varepsilon, \delta$, there exists $S$ in the commutant of $T$ such that $1\leq\|S\|<1+\delta$ and $\|Sx\|<\varepsilon$.
This is true for some $T$ (for example the identity), but is it true for all $T$? Is there anything known in this direction, even for $\ell_2$?