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 $m_x(T)x = 0$