1. INTRODUCTION

3

belonging to a noncommutative version S(W, W*) of the Schur-class. As before, this

S G S(W1 W*) is a complete unitary invariant for such Cuntz scattering systems.

In our companion paper [BaV04] we obtained a classification of isometric

representations of the Cuntz algebra where one in addition specifies a choice of

cyclic subspace £. The classification is in terms of a block matrix [W^^c*,^] with

rows and columns indexed by the Td x Td (where Td is the free semigroup on d

letters) possessing a combination of Toeplitz and Hankel properties; for this reason

we shall call such matrices "Haplitz matrices". In another report [BaV03], we

identify this model as an example of a noncommutative formal reproducing kernel

Hilbert space, a Hilbert space of formal power series in finitely many noncommuting

variables having many properties analogous to classical reproducing kernel Hilbert

spaces. This is one of the main tools for the discussion here, and will be reviewed

(along with some additional needed results) in the preliminary Chapter 2.

The Cuntz analogue of a unitary colligation is as follows. A d-variable unitary

colligation is a unitary operator of the form

(1.2)

U =

'A

B

C D

=

'A!

Ad

B{

Bd

D

-+

'@pi~

The positive time axis Z + is replaced by a free semigroup Td with generators equal

to the d letters gi,...,g

d

, and the associated J^-time system is given by

x(giw) = Aix(w) + Biu{w)

(1.3)

S:

x(gdw) = Adx(w) + Bdu(w)

y(w) = Cx(w) + Du{w)

where the variable w — gin ... g^ is a word in the symbols g\,...,gd, i.e., a generic

element of the free semigroup Td- The Fourier transform in this context we take

to be x(w) ^ x(z) =

^2weJrdx(w)zw

where

zw

= zin ... zix if w = gin ... gi±

where z — {z\,..., zd) is a d-tuple of formal, noncommuting indeterminants. Upon

application of this generalized Fourier transform to the system equations (1.3) and

under the assumption that the state of the system is initialized at 0 (so x(0) = 0

where 0 is the empty word, the unit element of T&), we arrive at the input-output

relation

y(z) = Wz(z)u(z)

where the transfer function in this case is given by

Tx(z) = D + C(I-

Zr(z)AYxZr(z)B

where we have set Zr(z) equal to the block-row matrix function

Zr(z)=[z1In ... zdIn].

To make connections with the scattering framework, it is also necessary to run the

system in backwards time; this is more complicated and is described in Chapter 3.

We mention that such formal power series have come up in various contexts in the

system theory literature (see e.g. [F174, BeOl, BeD99]).

There is also a dilation and operator model theory which fits into this set-

ting. Rather than considering a single contraction operator T on a Hilbert space