Condition for a sum of more than two projection operators in a Hilbert space to be a projection operator

Question

Let $H$ be an infinite-dimensional Hilbert space, and $P_1, \dots, P_m$ be a finite set of projection operators that project $H$ onto subspaces $Y_1, \dots, Y_m \subseteq H$, respectively. Define $P := P_1 + \cdots + P_m$. We know that in the simple case $m=2$, $P$ is a projection operator if and only if $Y_1 \perp Y_2$. My question is, what can we say for $m>2$? Is it still a necessary condition that $Y_i \perp Y_j$ for all $i \ne j$?

Answer 1

For $v\in Y_1$ we have $$\|v\|^2\ge \langle (P_1+P_2+\ldots +P_m)v,v\rangle \\ =\|v\|^2+\langle P_2v,v\rangle +\ldots +\langle P_mv,v\rangle \ge \|v\|^2$$ Hence $$\langle P_2v,v\rangle=\ldots =\langle P_mv,v\rangle=0$$ Therefore $P_2v=\ldots=P_mv=0$ which is equivalent to $Y_1\perp Y_2,\ldots, Y_m.$ By changing the order of projections we get $Y_i\perp Y_j$ for $i\neq j.$