$S$ and $T$ are idempotent linear operators in $V$ which is a vector space in $\mathbb{C}$. Prove that if $S+T$ is idempotent then $ST=TS=0$.

Question

If $S+T$ is idempotent, then $$(S+T)^2=S+T$$ $$\implies S^2+ST+TS+T^2=S+T$$ $$\implies S+T+ST+TS=S+T$$ $$\implies ST+TS=0$$

Now, from here how do I show that $ST=TS=0$?

Answer 1

We have $$ ST + TS = 0 \implies\\ S[ST + TS] = 0 \implies\\ ST + STS = 0. $$ Similarly, we have $TS + STS = 0$. Putting these results together yields $$ ST = -STS = TS. $$ That is, we indeed have $ST = TS$. Since $ST + TS = 0$, conclude now that $ST = 0$.

Answer 2

Essentially equivalent answer but more elegant solution: $ST = SST = -STS = TSS = TS$, so $2ST = ST+TS = 0$ and hence $ST = 0$. Note that you do not need the field $F$ underlying the vector space to be $ℂ$, but you do need $\text{char}(F) ≠ 2$ otherwise you cannot divide by $2$.