UNITARY DILATION S O F HILBER T SPAC E OPERATOR S 3 and henc e Tnh= P^V n h fo r h e § an d « = 0 , 1 , ••• . Thus U is a unitar y dilatio n o f T. Next observ e that , fo r an y h Sp , UA- 7 A = l / ^ . : . , 0 , [U , 0 , ^ - ( , 0 , [TH , 0 , . . . ^ = (••, o , HE,O A ,o,.-.\-/..., o , \fg 9 o , •-^ and, b y analogou s reasoning , l/*A-T* A = / . . - , 0 , D ^ , H , 0 , . . . \ therefore, th e subspace s (1.2.7) Q = (U-T)$? an d C * = ( l / * - T * ) $ consist respectivel y o f th e vector s ( •, 0 , \o\, d, 0 , •) wit h arbitrar y d e 2) , an d of th e vector s ( •, 0 , d^, \o\, 0 , ••) wit h arbitrar y d^ S % . I t the n follow s that , for n 0 , ^ = l ( ' , ' J 0 , i , ? , . . . , 0 ) T o ) - V / e ® an d u**C* = { ( ., o , "2£T o,...,^7[ol, o,.-\ : * * e ®*} . Bearing als o i n min d th e wa y S p wa s embedde d i n ft w e conclud e tha t ft ha s th e fol - lowing decomposition : (1.2.8) ft=... e ir 2 Q* e i r ! Q* eG* e £ e Q e u G e u 2 G e ... . From (1.2.7 ) an d (1.2.8 ) i t follow s readil y tha t th e minimalit y conditio n (1.2.3 ) i s als o fulfilled. It remain s t o prov e uniqueness . Le t U an d U b e an y tw o minima l unitar y dila - tions o f T , sa y o n ft' an d ft", an d observ e firs t tha t (U'nhv U' m h2) = (U'n-mhv h 2 ) = (Tn-mhv b 2 ) = (u" n ~mhv b 2 ) = {u"nhv u" m h2) for an y ^. , L e § an d fo r n m\ equalit y o f th e tw o extrem e member s holds b y th e 1 Not e tha t i n thi s notatio n th e sta r i n £ doe s not mea n adjugation : i t i s use d onl y t o indicate symmetr y i n th e definition s o f tf an d £ . http://dx.doi.org/10.1090/cbms/019/01
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