1. Isometri c an d Unitar y Dilation s o f a Contractio n Operato r 1. I n th e firs t tw o section s th e Hilber t space s whic h w e conside r ar e eithe r al l real o r al l complex the y ma y b e eithe r separabl e o r not . Operator s ar e alway s linea r an d bounded. Let A an d B b e operator s o n th e Hilber t space s 2 1 an d B , respectively . W e cal l B a dilation o f A (also , a "strong " o r "power " dilation ) i f 8 1 is a subspac e o f B an d the followin g conditio n i s satisfied : (1.1.1) A n = P % Bn\U ( n = 0 , 1 , ), Pu denotin g orthogona l projectio n fro m B ont o 21 . Conditio n (1.1.1 ) i s equivalen t t o the conditio n (1.1.2) (A n hv b 2 ) = (B"b v h 2 ) (h v h 2 « n = 0 , 1 , 2 , •) hence i f B i s a dilatio n o f A , the n B i s a dilatio n o f A . Condition (1.1.1 ) i s satisfie d i n particula r i f (1.1.3) AP % ~ P % B holds (o n B) indeed , (1.1.3 ) implie s A n P% = Pyfi n ^ or n = 0, 1 , •, an d restrictio n of bot h side s t o 2 1 yield s (1.1.1) . Conversely , (1.1.1 ) implie s AP%(Bna) = AA n a= A n+l a = P % Bn*la= P % B(Bna) for a 2 1 an d n = 0 , 1 , henc e (1.1.3 ) hold s whe n bot h side s ar e restricte d t o th e subspace V ^ _ 0 ^ " ^ ^ ^) - Thu s i f thi s subspac e equal s B (i.e. , i f B ha s n o prope r subspace containin g 2 1 an d invarian t fo r B) , the n condition s (1.1.1 ) an d (1.1.3 ) ar e equivalent. The importanc e o f th e abov e notio n come s fro m th e fac t tha t a n operato r A o f general typ e ma y hav e a dilatio n B o f som e quit e specia l type , an d henc e th e stud y o f A ca n b e reduce d t o th e stud y o f thi s B o f specia l type . 2. Fo r a contractio n operato r T on th e Hilber t spac e § (i.e. , wit h ||T| | l ) w e define th e defect operator D, defect space 2 ) and defect index b b y D=(/-T*T) 1/2 , ® = 5f, b = dim®, 1
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