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 nhv 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 nP% = 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 na= A n+la = 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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