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Math Help - Partial Isometries

  1. #1
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    Question Partial Isometries

    In Theorem 2.17

    The first half of the proof is:

    A course in functional analysis - Google Book Search

    Then the second half which is not shown:

    Now let B be a closed symmetric extension of A. By Lemma 2.15 there is an A symmetric, A-closed manifold M in L_+ + L_- such that gra B = gra A + gra(A* | M ). If f in M, let f = f^+ + f^-, where f^+/- in L_+/-; put I_+ = {f^+: f in M}.
    Since M is A-symmetric, 0 = <A*f,f>-<f,A*f>= 2i<f+,f+>-2i<f^-,f^->; hence ||f^+ || = || f^-|| for all f in M. So if Wf^+ = f^- whenever f = f^+ + f^- in M and if I_+ is closed, W is a partial isometry and (2.18) and (2.19) are easily seen to hold.

    Why is the last statement true?
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  2. #2
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    ||f^+||=||f^-||, we have that
    ||W f^+||=||f^-||=||f^+||
    so you need only show that the domain of W, I_+, is closed.

    The part g\oplus A^*g\in cl{gra(A^*|M)} is by the fact that it is a limit
    point and gra(A^*\vert{M}) is closed since M is an A-closed manifold,
    so g\oplus A^*g\in gra(A^*|M). Thus, g\in M. Hence, g^+\in I_+
    and so I_+ is closed.
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