10 2. FUNCTIONAL MODELS
as formal series f(z,C) J2V w^Td*Td fv,wZvCw in the space Cw- It is natural to
define a Fourier representation operator I by
(2.2.9) $ : / c ^ Y^
{PsUwWvk)zvC-
v,w$zTd
Then one can show that $ is a coisometry mapping /C onto Cw- Moreover there is
a model row unitary d-tuple Uw (M\v,i Mw,d) on Cw such that
(2.2.10) QUj = Uwtj$, §U] = U^jQ for j = 1,... d,
and we recover W from $ via
(2.2.11) W = $ $
w
and Cw = $ /C.
The precise formula for Uw (at least on the dense subset WV{Td x Td,£)) is
(2.2.12) WWjJ-: Wp .- WSpp for p G P(.Fd x
J ^ , £ ) .
where Sp is the operator of multiplication by the indeterminate Zj on the right:
(2.2.13) Sf:f(z,C)^f(z,C)-zJ,
and can be viewed as a noncommutative version of a unilateral shift. The adjoint
l^wj °f Mw,j is given densely by
(2.2.14) Ufa : Wp h-
WUf[*]p
where Up is a noncommutative version of a bilateral shift:
(2.2.15) Uf: f(z, 0 ~ /(0, C) C"1 + /(*, 0 zj
where
/(o,0= E h,*CXf(z,0= E /f.u-^C"-
w£Td v.wE^d
and £/• w denotes the formal adjoint of Up with respect to the L2-inner product,
given by
- i
3 '
(2.2.16) Uf["]: f(z, C) - /(0, C) 0 + /(*, 0 *
For the record, the formal adjoint S- of Sp with respect to the
L2-inner
product
is given by
(2.2.17)
SfM:f(z,C)^f(z,C)-z-1.
Here the same convention with respect to expressions of the form
zv

zw
(and
hence, in particular, for zv zj1 = zv Z9J ) as was used for expressions of the
form UVUW is in place (see (2.2.1) and the explanation there). In particular,
if £ is *-cyclic for the row-unitary U {U\,... Md), then $ implements a unitary
equivalence between the row-unitary U on /C and the model row unitary Uw on Cw-
It turns out that the Haplitz property (2.2.3)-(2.2.4) imposed on a block-operator
matrix W is equivalent to the intertwining condition
(2.2.18) WSf = UpW on V{Td x Td, £).
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