operators that commute with idempotent operators If E is an idempotent operator on a finite dimensional vector space say V and T is another linear operator on the same space V that commutes with E i.e. TE=ET, then what can said about T?

You can check that $T$ commutes with $E$ if and only if you can write
$$
 T = ETE + (1-E)T(1-E),
$$
which is that same thing as saying that $T$ is block-diagonal with respect to the direct sum decomposition $V = \operatorname{im}(E) \oplus \operatorname{im}(1-E)$, i.e.,
$$
 T = \begin{pmatrix} T_1 & 0 \\ 0 & T_2 \end{pmatrix},
$$
where $T_1 : \operatorname{im}(E) \to \operatorname{im}(E)$ and $T_2 : \operatorname{im}(1-E) \to \operatorname{im}(1-E)$. This is really the most that you can say about the structure of $T$ commuting with $E$. In terms of invariant subspaces, you could equivalently say that commutes with $E$ if and only if it leaves $\operatorname{im}(E)$ invariant and leaves $\operatorname{im}(1-E)$ invariant.
