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Math Help - Self-Adjoint Projections

  1. #1
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    Self-Adjoint Projections

    Problem: P_1 and P_2 are two self-adjoint projections. Is it true that \mbox{R} (P_1) \subseteq \mbox{R} (P_2) \Longrightarrow P_1 \leq P_2? (Note that R() indicates the range).

    I believe it is not true, but I cannot find any counterexamples or way to disprove.
    One fact that might be useful to consider is that for a self-adjoint projection \mbox{R}(P) = \mbox{N}(P)^\perp
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  2. #2
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    Quote Originally Posted by DJDorianGray View Post
    Problem: P_1 and P_2 are two self-adjoint projections. Is it true that \mbox{R} (P_1) \subseteq \mbox{R} (P_2) \Longrightarrow P_1 \leq P_2? (Note that R() indicates the range).

    I believe it is not true, but I cannot find any counterexamples or way to disprove.
    One fact that might be useful to consider is that for a self-adjoint projection \mbox{R}(P) = \mbox{N}(P)^\perp
    If \mbox{R} (P_1) \subseteq \mbox{R} (P_2) then P_2P_1 = P_1. Taking adjoints, you see that P_1P_2= P_1. Thus P_1 and P_2 commute, and it follows (as you can easily check) that (P_2 - P_1)^2 = P_2 - P_1. Thus P_2 - P_1 is also a selfadjoint projection and is therefore positive. Hence P_1 \leqslant P_2.
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