UNITARY DILATION S OF HILBER T SPAC E OPERATOR S 5
oo
(1.4.1) ®
+
= V U%%.
0
Therefore, recallin g (1.1.3) w e hav e
(1.4.2) TP
+
= P
+
U +
where P
+
denote s th e orthogona l projectio n o f S
+
ont o § .
One show s i n analog y t o th e unitar y cas e tha t isometri c dilation s o f T satisfyin g
this kin d o f minimalit y conditio n ar e isometricall y isomorphi c s o tha t th e minima l iso -
metric dilatio n o f T i s (essentially ) unique.
Note tha t i f w e star t wit h th e representatio n o f $ an d U a s give n b y formula s
(1.2.4) an d (1.2.5) the n S
+
wil l consis t o f th e vector s k ft wit h zer o component s
h (n - 1 ) s o that , b y a natura l embedding , ft
+
ca n b e considere d a s th e spac e o f
vectors
k= (h
Q
, h
v
h
2
, .«* ) wit h h
Q
e jg , h
n
e % in l) ,
and \\k\\ = ( 2 ~ | | ^ | | 2 ) 1 / 2, an d V
+
is the n define d b y
(1.4.3) U+k= (Th
Q
, Dh
Qt
h
v
b
2
, - .
It i s a n eas y exercis e t o verif y directl y tha t thi s operato r i s a minima l isometri c dila -
tion o f T .
In th e seque l w e shal l no t kee p t o a particula r realizatio n o f th e minima l isometri c
dilation (/
+
o f T , bu t w e shal l alway s conside r i t a s th e restrictio n o f th e minima l
unitary dilatio n U. W e shal l hav e
(1.4.4) ®
+
=:£©Af
+
(Q) .
Let u s no w conside r th e Wol d decompositio n o f S
+
correspondin g t o th e isometr y
t/
+
, i.e. , th e decompositio n
(1.4.5) ®
+
= M
+
( S * ) © & ,
where
oo
(1.4.6) Q * = S
+
© ^ + ®+ an d K = f l l/^*
+
.
o
The Wol d decompositio n (1.4.4) reduces th e operato r t /
+
; l/
+
i s a unilatera l shif t o n
M+(Q) an d unitar y o n Si . Fo r th e wanderin g subspac e C * generatin g th e unilatera l
shift par t o f U
+
w e obtain , usin g th e equalitie s R
+
= § © Af
+
(£) = 5 ? © C © U
+
M+(C)
and l /
+
ft
+
= U
+
( § © M
+
(G)) = l /
+
lg © l /
+
M+ (8), tha t
G* = ( § © Q ) 0 ( /
+
§ .
http://dx.doi.org/10.1090/cbms/019/02
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