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
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