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$

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