4
BEL A SZ.-NAGY
the symmetr y propert y o f th e inne r product—i n cas e n m also . Henc e w e conclud e
that th e ma p
OO OO
£ V' nhn M £ U" nhn (h
n
e § ; h
n
= Q fo r |n | larg e enough )
oo
—oo
is isometric , an d therefor e extend s b y continuit y t o a n isometr y cf o f ft' ont o ft".
This isometr y leave s th e vector s i n § invariant , an d carrie s U' int o (/" , i.e. , cfU' =
U* f. Thu s i f w e disregar d suc h isometri c isomorphism s cf
y
th e minima l unitar y dila -
tion o f T is unique .
In th e seque l on e may , bu t nee d not , restric t onesel f t o an y specia l realizatio n o f
the minima l unitar y dilatio n U o f T . Th e subspace s 0 an d Q a s define d b y (1.2.7)
are wandering fo r U an d th e decompositio n (1.2.8) holds , independentl y o f th e spe -
cial realizatio n o f ft an d U .
3. Th e followin g equalit y i s o f importance :
(i.3.i) § ec= u § © uc*.
It suffice s t o prov e tha t
(1.3.2) £ © ( I / - T ) § - t/j g e ( J - l / r * ) § .
The orthogonalit y o f (7 § an d (7 8 follow s fro m th e orthogonalit y o f § an d Q (se e
(1.2.8)), an d equalit y (1.3.2) follow s fro m th e fac t tha t ever y vecto r k o f ft whic h ca n
be writte n i n on e o f th e form s
k= Uh^il-. UT*)h
2
(b
v
h2ۤ)
can b e writte n i n th e othe r also : on e ha s onl y t o se t
i1
=
rV + (/- r*7W,A ' = r£1+ (/-rr*U2,
o r
A2= ti -Tti\ h" = ij-T*^ .
4. On e o f th e consequence s o f (1.3.1) i s tha t U § C § $ 2 , whenc e U" § C § ©
C © . © (/* " * G U = 0 , 1, - 0. O n th e othe r hand , w e clearl y hav e C C § V (7§ ,
whence U* Q C U" § V U n* l % (n = 0 , 1, X W e conclud e tha t th e subspace s £ ©
M+(l/; Q ) an d V ^ ^ " § ° ^ ^ a r e equal . Denot e thi s subspac e b y ft
+
an d se t U
+
=
[j|ft
+
. Clearl y U
+
i s a n isometric dilatio n o f T , indee d a minimal one , tha t is ,
satisfying
2 A subspac e * o f th e spac e of a n isometr y V i s calle d "wandering " i f %, V%, V U , ••
are mutually orthogonal . The n w e denote th e orthogonal su m ©
0
V n% b y M
+
( F; H); i f ther e i s
no ambiguity , w e als o writ e M
+
(H). I n cas e V i s unitary , V - V, w e writ e M(£/ ; 31) or M(fc )
for ©° ° U n%
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