UNITARY DILATION S OF HILBER T SPAC E OPERATOR S 5 oo (1.4.1) ® + = V U%%. 0 Therefore, recallin g (1.1.3 ) w e hav e (1.4.2) TP + = P + U + where P + denote s th e orthogona l projectio n o f S + ont o § . One show s i n analog y t o th e unitar y cas e tha t isometri c dilation s o f T satisfyin g this kin d o f minimalit y conditio n ar e isometricall y isomorphi c s o tha t th e minima l iso - metric dilatio n o f T i s (essentially ) unique. Note tha t i f w e star t wit h th e representatio n o f $ an d U a s give n b y formula s (1.2.4) an d (1.2.5 ) the n S + wil l consis t o f th e vector s k ft wit h zer o component s h (n - 1) s o that , b y a natura l embedding , ft + ca n b e considere d a s th e spac e o f vectors k= (h Q , h v h 2 , .«* ) wit h h Q e jg , h n e % in l) , and \\k\\ = ( 2 ~ | | ^ | | 2 ) 1 / 2 , an d V + is the n define d b y (1.4.3) U+k= (Th Q , Dh Qt h v b 2 , - . It i s a n eas y exercis e t o verif y directl y tha t thi s operato r i s a minima l isometri c dila - tion o f T . In th e seque l w e shal l no t kee p t o a particula r realizatio n o f th e minima l isometri c dilation (/ + o f T , bu t w e shal l alway s conside r i t a s th e restrictio n o f th e minima l unitary dilatio n U. W e shal l hav e (1.4.4) ® + =:£©Af + (Q) . Let u s no w conside r th e Wol d decompositio n o f S + correspondin g t o th e isometr y t/ + , i.e. , th e decompositio n (1.4.5) ® + = M + ( S * ) © & , where oo (1.4.6) Q * = S + © ^ + ®+ an d K = f l l/^* + . o The Wol d decompositio n (1.4.4 ) reduces th e operato r t / + l/ + i s a unilatera l shif t o n M+(Q) an d unitar y o n Si . Fo r th e wanderin g subspac e C * generatin g th e unilatera l shift par t o f U + w e obtain , usin g th e equalitie s R + = § © Af + (£) = 5 ? © C © U + M+(C) and l / + ft + = U + ( § © M + (G)) = l / + lg © l / + M+(8), tha t G* = ( § © Q ) 0 ( / + § . http://dx.doi.org/10.1090/cbms/019/02
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