conductor of a vector

Let $k$ be a field, $V$ a vector space, $T:V\to V$ a linear transformation, and $W$ a $T$-invariant subspace of $V$. Let $x\in V$. The $T$-conductor of $x$ in $W$ is the set $S_T(x,W)$ containing all polynomials $g\in k[X]$ such that $g(T)x\in W$. It happens to be that this set is an ideal of the polynomial ring. We also use the term $T$-conductor of $x$ in $W$ to refer to the generator of such ideal.

In the special case $W=\{0\}$, the $T$-conductor is called $T$-annihilator of $x$.Another way to define the $T$-conductor of $x$ in $W$ is by saying that it is a monic polynomial $p$ of lowest degree such that $p(T)x\in W$. Of course this polynomial happens to be unique. So the $T$-annihilator of $x$ is the monic polynomial $m_x$ of lowest degree such that $m_x(T)x=0$.