4 BEL A SZ.-NAGY the symmetr y propert y o f th e inne r product—i n cas e n m also . Henc e w e conclud e that th e ma p OO OO £ V' n hn M £ U" n hn (h n e § h n = Q fo r |n | larg e enough ) oo —oo is isometric , an d therefor e extend s b y continuit y t o a n isometr y cf o f ft' ont o ft". This isometr y leave s th e vector s i n § invariant , an d carrie s U' int o (/" , i.e. , cfU' = U* f. Thu s i f w e disregar d suc h isometri c isomorphism s cf y th e minima l unitar y dila - tion o f T is unique . In th e seque l on e may , bu t nee d not , restric t onesel f t o an y specia l realizatio n o f the minima l unitar y dilatio n U o f T . Th e subspace s 0 an d Q a s define d b y (1.2.7 ) are wandering fo r U an d th e decompositio n (1.2.8 ) holds , independentl y o f th e spe - cial realizatio n o f ft an d U . 3. Th e followin g equalit y i s o f importance : (i.3.i) § ec= u § © uc*. It suffice s t o prov e tha t (1.3.2) £ © ( I / - T ) § - t/j g e ( J - l / r * ) § . The orthogonalit y o f (7 § an d (7 8 follow s fro m th e orthogonalit y o f § an d Q (se e (1.2.8)), an d equalit y (1.3.2 ) follow s fro m th e fac t tha t ever y vecto r k o f ft whic h ca n be writte n i n on e o f th e form s k= Uh^il-. UT*)h 2 (b v h2€§) can b e writte n i n th e othe r also : on e ha s onl y t o se t i 1 = rV + (/- r*7W,A ' = 1 + (/-rr*U 2 , o r A2= ti -Tti\ h" = ij-T*^ . 4. On e o f th e consequence s o f (1.3.1 ) i s tha t U § C § $ 2 , whenc e U" § C § © C © . © (/* " * G U = 0 , 1 , - 0. O n th e othe r hand , w e clearl y hav e C C § V (7§ , whence U* Q C U" § V U n * l % (n = 0 , 1 , X W e conclud e tha t th e subspace s £ © M+(l/ Q ) an d V ^ ^ " § ° ^ ^ a r e equal . Denot e thi s subspac e b y ft + an d se t U + = [j|ft + . Clearl y U + i s a n isometric dilatio n o f T , indee d a minimal one , tha t is , satisfying 2 A subspac e * o f th e spac e of a n isometr y V i s calle d "wandering " i f %, V%, V U , •• are mutually orthogonal . The n w e denote th e orthogonal su m © 0 V n % b y M + ( F H) i f ther e i s no ambiguity , w e als o writ e M + (H). I n cas e V i s unitary , V - V, w e writ e M(£/ 31 ) or M(fc ) for ©° ° U n %
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