I am trying to prove the following equivalences.
Let $P_1$ and $P_2$ be the orthogonal projections on the closed subspaces $M_1$ and $M_2$ respectively. The following are equivalent:
(i) $P_1+P_2$ is an orthogonal projection
(ii) $P_1P_2 = 0$
(iii) $M_1\perp M_2$
I have proved $(ii) \to (iii)$, any hint on the other implications would be precious.
Best Answer
Let me prove (i) $\Rightarrow$ (ii), while giving you some more time to think about (iii) $\Rightarrow$ (i), which is not so hard.
By hypothesis $$ P_1+P_2= (P_1+P_2)^2 = P_1^2+P_1P_2+P_2P_1+P_2^2 = P_1+P_1P_2+P_2P_1+P_2. $$ so $P_2P_1=-P_1P_2$. Left multiplying the above by $P_2$ we get $$ P_2P_1=-P_2P_1P_2. $$ Since $P_2P_1P_2$ is self-adjoint, then so is $P_2P_1$. But then $$ P_1P_2= P_1^*P_2^*= (P_2P_1)^* = P_2P_1 = -P_1P_2, $$ whence $P_1P_2=0$.