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 $\ell\_2$?



Operator in the commutant which is small on a given vector
Not even true for $2\times 2$ matrices. Let $T$ be the nilpotent $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ and $x$ be the vector $(0,1)$. Then anything in the commutant of $T$ has form $\lambda+\mu T$. So if $S$ is in the commutant of $T$, then $\|Sx\| = \sqrt{\lambda^2 + \mu^2}$, so if $\|Sx\| < \varepsilon$, then $\max(|\lambda|, |\mu|) < \varepsilon$ and $\|S\| < \sqrt{\frac {3 + \sqrt 5} 2}\varepsilon$.


