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Let $\mathcal{H}$ be a Hilbert space and let $P$ and $Q$ be two orthogonal projections to closed subspaces $M$ and $N$ respectively.

Prove that:

  • If $PQ$ is an orthogonal projection then it's range is $M\cap N$
  • $PQ$ is orthogonal iff $PQ = QP$

I got stuck with the first clause, any hint would be most welcomed.

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If $PQ$ is an orthogonal projection then in particular $PQ = (PQ)^\ast$, hence $$PQ = (PQ)^\ast = Q^\ast P^\ast = QP$$ because $P = P^2 = P^\ast$ and $Q = Q^2 = Q^\ast$ by hypothesis. Thus $M \ni PQx = QPx \in N$ and hence $PQ(\mathcal{H}) \subset M \cap N$. For all $x \in M \cap N$ we have $x = Px$ and $x = Qx$, hence also $x = PQx$, so $PQ(\mathcal{H}) = M \cap N$. This shows the first claim as well as one direction of the second assertion.

For the other direction verify that $PQ = (PQ)^2 = (PQ)^\ast$.

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