By Per-Olov Lowdin
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The self-adjoint character of A follows from because We have thus shown that every finite resolution of the identity generates a bounded, self-adjoint transformation in Hilbert space. If the region of variability of Ed is confined to the interval - K to + K , then Having proved the self-adjoint character of A , we adopt for this operator the symbol H which is customary for self-adjoint operators. The operator H is permutable with every element of the family which forms the resolution of the identity.
36) Therefore DBt 3 D, and Btg = xg = Bg. The operator B is symmetric. To prove that B is self-adjoint we must show that DBt = D,. 54 Bela A. Lengyel Letfe 2 2 ,and let f be so restricted that f ( x ) = 0 when 1x1 exceeds some large number N. Then f e DB, and we may write for every g in DBt and h = B'g that fr,f(x)xg(x)* dx - J:_f(x)h*(x) dx = JYN f(x)[xg(x) - h(x)]* dx = 0 . 37) Since this equation holds for any truncated functionf(x), it follows that whenever 1x1 < N , then h(x) = xg(x). &. Thus g , which was an arbitrary element of D,t , belongs to DB.
Moreover, we found that this decomposition is unique. 35) defines the projection operator P associated with M. The domain of P is 5, its range is M. 34) by complex constants and arriving at af+ Pg = (afo + Pgo) + (4+ PSI), (TI. 36) where the first expression in parenthesis belongs to M, the second to MI. Every projection is a bounded operator with the bound 1. Moreover P 2 = P,because Pfo =fo. For arbitrary f and g we have (PAg) = (Yo, g) Conversely, given a bounded, self-adjoint operator P,such that P2 = P, 27 Functional Analysis for Quantum Theorists there is always a closed linear manifold M of which P is the projection operator.