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Math Help - Projection

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    Projection

    Let H be a Hilbert Space and P and Q be the projections on closed linear subspaces M and N of H.
    Prove that if Q-P is a projection,then its range is N\cap M^\perp.
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    Quote Originally Posted by problem View Post
    Let H be a Hilbert Space and P and Q be the projections on closed linear subspaces M and N of H.
    Prove that if Q-P is a projection,then its range is N\cap M^\perp.
    For a start, I-P is a projection, with range M^\perp. So if x\in N\cap M^\perp then (I-P)x=x, Px=0, Qx=x and hence (P-Q)x=x. Thus if P-Q is a projection then its range contains N\cap M^\perp.

    For the converse inclusion, if P-Q is a projection then
    (1)\quad (Q-P)^2 = Q-P,
    (2)\quad Q - PQ - QP + P = Q-P (since P^2=P and Q^2 = Q),
    (3)\quad PQ+QP=2P.
    Multiply both sides of (3) on the left to get
    (4)\quad QPQ=QP.
    Multiply both sides of (3) on the right to get
    (5)\quad QPQ=PQ.
    From (4), (5) and (3),
    (6)\quad PQ=QP=P.
    Therefore
    (7)\quad (I-P)Q=Q(I-P)=Q-P.
    It follows from (7) that the range of Q-P is contained in the range of (I-P) and in the range of Q, and thus in N\cap M^\perp.
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