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Let $H$ be a separable Hilbert space. Can you provide an example of 3 orthogonal projection operators which are mutually non-commuting?

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  • $\begingroup$ Hint: If $PQ = QP$, then what can you say about the subspaces $P(H)$ and $Q(H)$? $\endgroup$ Oct 19, 2013 at 8:54
  • $\begingroup$ If $PQ = QP$, then this map is the projection onto the subspace $P(H) \cap Q(H)$. This scenario makes sense. I have a harder time visualizing what happens if $PQ \ne QP$. $\endgroup$ Oct 19, 2013 at 9:23
  • $\begingroup$ why do you need three projections? $\endgroup$
    – Noix07
    Mar 14, 2014 at 16:23

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Take three distinct unit vectors which are not pairwise orthogonal, and consider the orthogonal projections on the lines they span.

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  • $\begingroup$ and the product of two such projections is NOT the intersection of the images!!! (got confused for an hour by the comment of LaGatta) $\endgroup$
    – Noix07
    Mar 14, 2014 at 17:49

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