Let $P_{1},P_{2}$ be orthogonal projectors on finite unitary space. With what condition is $P_{1} + P_{2}$ an orthogonal projector and on which space is it projecting?
I have concluded that the condition is $P_{1} * P_{2} = - P_{2} * P_{1}$. For the space it is projecting, I believe I should observe $Im (P_{1} + P_{2})$ but since $Im (P_{1} + P_{2}) \subseteq Im (P_{1}) + Im(P_{2}) $ I don't see how to conclude anything from there. Any hint helps!
Note that $P_1+P_2$ will always be self adjoint. Hence $P_1+P_2$ is a projection precisely when $(P_1+P_2)^2=P_1+P_2$, which occurs precisely when $P_1 P_2=-P_2 P_1$ (1). If this equality holds note that we can rewrite the operator $-P_1P_2P_1P_2$ as $-(-P_2P_1)P_1P_2=P_2P_1P_2$, but we can also rewrite it as $-P_1(-P_1P_2)P_2=P_1P_2$. Hence $P_1P_2=P_2P_1P_2$. Taking adjoints on both sides we get $P_2P_1=P_2P_1P_2$ so we see that actually $P_1P_2=P_2P_1$. In conjunction with (1) this clearly shows that $P_1P_2=P_2P_1=0$ so $P_1H$ and $P_2H$ must be mutually orthogonal subspaces. Conversely, if $P_1H$ and $P_2H$ are mutually orthogonal subspaces it's easy to see that (1) holds (both sides are then $0$).