Sum of orthogonal projection operators

Question

Let $X$ be an Hilbert space, I am trying to see that if $P$ and $Q$ are orthogonal projection operators then the following are equivalent:

$(1) Im Q\subseteq KerP$

$(2)P+Q$ is an orthogonal projection operator.

I was able to do $(1) \implies (2)$ but I cant seem to make the other one so any tips are aprecciated, what I tried revolves around this

From the condition that $P+Q$ is a projection operator I know that $PQ=-QP$ but I cant seem to see why this will have to be zero, I dont know if looking at the decompositions of the space $X$ would be righ way to look at this, also tried using the fact that the operator is positive but got nowhere.

Also another thing I am trying to see is that $Im(P+Q)$ is the closure of the subspace generated by $Im P$ and $Im Q$, and any advice here would be aprecciated too, I was able to show that $cl(span\{ImP+ImQ\})\subseteq Im(P+Q)$.

Thanks in advance !

Answer 1

Let us take $f\in Im(Q)$ and hence $Qf=f$ because $Q$ is a projection. Now apply $P$ on both sides to obtain $$Pf=PQf=-QPf.$$

Apply $Q$ on both sides and use that $Q^2=Q$ to obtain $QPf=-QPf.$ Which gives $QPf=0$ and hence $PQf=0.$ Now again using the fact that $Qf=f$ we get $Pf=0$ which yields the claim.