$\mathbf {The \ Problem \ is}:$ Let, $V$ be a vector space over a field $\mathbb F$ . Determine all possible linear maps $T : V \to V$ such that $T^k = T$ for some $k \in \mathbb N$ .
$\mathbf {My \ approach} :$ Obviously, if $V$ is finite dimensional, then for $T^2=T$ ; for any vector $x \in V$ , $x= (T(x)) + (x - T(x))$ and so $V= \mathcal F \oplus Ker T$ where $\mathcal F = \{ y \in V \ | \ T(y)=y \}$ . In other words, $T$ represents all possible projection mappings of the subspace $\mathcal F$ along $Ker T$ .
Now, my question is can we always find some idempotent matrix in the sequence of degrees of $T$ when given $T^k =T$ ???
And what happens if $V$ becomes infinite dimensional ???
In the above two cases, is there any relation between $T$ and projection of a subspace along another subspace ???