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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 \begin{equation} S = \overline{span}\{S \cap Q^\perp, S \cap Q\}. \end{equation}

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

Thanks.

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