CHAPTER 2
Functional Models for
Row-Isometric/Row-Unitary Operator Tuples
2.1. The classical case
If /C is a Hilbert space, £ is a subspace of K and U is a unitary operator on /C,
we may associate a block-Toeplitz matrix [Wij]fj=_OQ by
WiJ=PeU*iUj\e=Pel(j-i\£-
Note that each matrix entry is an operator on £. It is convenient to view W as an
operator from the space V(Z, £) of trigonometric polynomials p(z) =
Yln=-NPnZn
with coefficients pn e £ into the space L(Z, £) of formal Laurent series f(z) =
J^^L-oo
fnZn
with coefficients fnE£ via the formula
oo TV
m = {wp)(z)iifi= E ^ f t = E
W^PJ-
J—

OC
j = N
From the factored form of Wij, it is easily seen that Wij is positive semidefinite
in the sense that
(Wp,p)L2 0 for all p e V(Z,£.
Here we are using that the usual L2-inner product
/ oo oo \ oo
( E /«*"' E
p«z")
= E (/-p«)£
\n= —oo n—
oo / ^2 n = 00
gives a well-defined pairing between L(Z,£) and P(Z,£), since the infinite sum for
such a pairing collapses to a finite sum. The image W V(Z, £) of this operator W
can be completed to a Hilbert space denoted by Cw with inner product defined by
(2.1.1) (WP,Wq)Cw = (Wp,q)L2.
As evaluation of Fourier coefficients f(z) i— fn is bounded in this norm, elements
of the completion can still be identified as formal series f(z) = X^^L-oo fnzU-
Moreover, the map £ given by
00
(2.1.2) $: k^ E
(PsK*nk)zn,
n =
—00
sometimes called the Fourier representation operator with respect to U and 5,
defines a coisometry from /C onto Cw such that
&U = Uw$
6
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