# Show that two projections commute if and only if their respectively closed subspaces are compatibles

I have the next problem:

Let $\mathcal{H}$ be a separable and complex Hilbert space, with $S$ and $Q$ two closed subspaces on it. Let $P_S$ and $P_Q$ be the orthogonal projection onto $S$ and $Q$ respectively.

We say that $S$ and $Q$ are compatibles if $$S = \overline{span}\{S \cap Q^\perp, S \cap Q\}.$$

I need to show that $P_S P_Q = P_Q P_S$ if and only if $S$ and $Q$ are compatibles.

Thanks.