UNITARY DILATION S O F HILBER T SPAC E OPERATOR S
3
and henc e
Tnh= P^V nh 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 , \fg9 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
2Q*
e i r
!
Q* eG* e £ e Q e u G e u
2G
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' mh2) = (U'n-mhv h
2
) = (Tn-mhv b
2
)
= (u" n~mhv b
2
) = {u"nhv u" mh2)
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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