, we have that
so you need only show that the domain of , , is closed.
The part is by the fact that it is a limit
point and is closed since is an A-closed manifold,
so . Thus, . Hence,
and so is closed.
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?