Composing Projections on a Hilbert Space

Question

Let $P,Q$ be projections on a Hilbert space such that $PQ$ is a projection.

I have been able to prove that $PQ=QP$.

I want to show that $ker(PQ)$ is contained in $ker(P)+ker(Q)$.

If there's a mathematician who would lend a hand, I'd be grateful.

Answer 1

Hint: For a projection $P$, we have

$$\ker P = \operatorname{im} (I - P).$$