A confustion about the proof of Primary Decomposition theorem

Question

In proofWiki, in the proof of Primary Decomposition Theorem while showing that (in the induction step)

$$\ker \left({p_i \left({T \restriction_W}\right)^{a_i} }\right) = \ker \left({p_i \left({T}\right)^{a_i} }\right),$$

how does the author concludes

$$p_i \left({T \restriction_W}\right)^{a_i} \left({v}\right) = p_i \left({T}\right)^{a_i} \left({v}\right) = 0$$

for $v \in \ker \left({p_j \left({T}\right)^{a_j} }\right)$ ?

Edit:

I'm referring to the proof that I have linked.

Answer 1

Since $v \in \ker \left({p_j \left({T}\right)^{a_j} }\right)$ implies $v \in W$, and when the input from the space $W$, both $T|_{W}$ and $T$ are the same, we should have

$$p_i \left({T \restriction_W}\right)^{a_i} \left({v}\right) = p_i \left({T}\right)^{a_i} \left({v}\right) = 0$$