By P. Fitzpatrick

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Example text

6n 1 1 C. a re m utually d isjoin t J A ls o , T h e re fo re and let 1 = { s | ||x (s ) - x || 0, F o r any n -1 +k x n n ), k the n and g has at m ost one 2 l|gn(s ) “ g (s )|| t g € L^(p; M = p ({ x | f(s ) ^ 0 } ) , and Thus f can be app roxim ated b y sim p le fun ction s. ”*■ 0 and w here f(s ) ^0 ), ||gn - g || 0. x . with x . in a fix e d countable dense subset and k. the J 3 3 J ch a r a c te r is tic function o f an open in terv a l with ration al endpoints; the set o f such functions is cou ntable.

35 3. F u nction s o f i -r— ds Since S o f the p reced in g s e ctio n is a s e lf-a d jo in t o p e r a to r, we can in p rin cip le define

R°° It e SU gx d s . -1 = R, 30 THEOREM 3. 1. adjoint o p e r a to rs {U j T h ere a re 1-1 c o rre s p o n d e n ce s betw een s e lf- S, s p e ctra l m ea su res {E ^ }, and unitary groups given b y : Et = e{ (S) , u t = ut (S) . S = / sdE g ; S = lim ¿ ( V i ) ; U = J e iSt dE . t ^ s P r o o f. If S is s e lf-a d jo in t and E^ = e^(S), then by uniqueness o f the extension s to S = f(S) = J 01 (IR) and the c o r o lla r y o f T h e o re m 2. 2, f(s )d E g, w here f(s ) = s. s p e ctra l m ea su re and r^ (s) = (\ - s) , g -*■ J*g (s)d E g.

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