Conductor

The set of polynomials $g$ that allow for the $g(T)$ of a linear transform $T$ to map values of a given $\alpha$ into a given set $W$.

Definition

• With $V$ as a $\mathbb{F}$ Vector Space
• With $T\in \mathcal{L}(V)$
• With $W$ as an Invariant Subspace for $T$
• With $\alpha \in V$
• Then, the $T$-conductor of $\alpha$ into $W$ is the set $S_T(\alpha ;W)=\{g\in \mathbb{F}[x]|g(T)\alpha \in W\}$

Intuition

• Every $g\in S_{\alpha }(\alpha ;W)$ will map values of $a$ into $W$

Corrolaries

• Conductors are Ideals

Concepts

• Annihilator