BELASZ.-NAGY
the ba r denotin g closure . Le t D^ , 5^. , b ^ hav e th e correspondin g meanin g fo r T .
The obviou s equatio n T(l - T*T) = T - TT*T = ( /- TT*)T implie s Tp(D 2) = p(D|) T
for an y polynomia l /?(A) , an d henc e fo r an y continuou s functio n o n 0 A 1, i n particu -
lar fo r th e functio n A . Thi s give s th e equation s
TD=D.T, DT* = T*D^
;
r^cSL f i c S .
(1.2.1)
whence w e deduc e
(1.2.2)
Next w e prov e
Theorem 1. Every contraction operator T on a Hilbert space § has a unitary di~
lation U on a Hilbert space ft. One can require that this unitary dilation be minimal
in the sense
(1.2.3)
V U

= ft;
U is then determined by T uniquely {that is, up to an isometric isomorphism leaving
the vectors of § invariant).
Proof. For m th e Hilber t spac e ft oi vector s
(1.2.4) * = /••-, h_
v
h_
v
g j , h
v
b
v
- ^
with component s h
Q
e § , h
n
e 2) , h_ e 3 * fo r « 1, an d nor m
1/2
IWI
= Z H*
OO,
and embe d § i n ft b y identifyin g h e S p wit h y •• , 0 , ]T] , 0 , y - Th e orthogona l
projection P ^ i s the n give n b y P ^ = \* •, 0 , l&J , 0 , \ = &
Q
. Defin e o n ft th e
operators V, U' b y
(1.2.5) W :
and
(
' ' ^ _ 3 ' ^ _ 2 '
D * * ^ + Th
Q
, - T ib_
1
+ Dh
0, i
r
4 2, •- ^
(1.2.6) ( A = / . -, 4 _
r
D ^
Q
- T i j .
T h
Q
+ Dh
x
, &
2
, ^3,
)
these definition s ar e correc t becaus e o f (1.2.2). Immediat e computation s base d o n th e
relations (1.2.1) yiel d tha t U is isometri c an d UU' = 1$; henc e w e conclud e tha t V is
unitary an d U' = U~ = U .
From (1.2.5) i t readil y follow s tha t
Unh = / . . . , 0 ,
Wnh\,
DT
n~lh,
... , Dh, 0, - \
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