Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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 $PQ = (PQ)^2 = (PQ)^\ast$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.