1. ## Partial Isometries

In Theorem 2.17

The first half of the proof is:

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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?

2. $||f^+||=||f^-||$, 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.