2

1. INTRODUCTION

to which corresponds a discrete-time, unitary, linear system

(

, v / x(n+l) = Ax(n) + Bu(n)

1 j \ y(n) = Cx(n) + Du(n).

Here the "time-variable" n is an element of the nonnegative integers Z

+

(or some-

times the set of all integers Z), the vector x(n) takes values in the state space 7Y, the

vector u(n) takes values in the input space £, and the vector y(n) takes values in the

output space £*. Application of the Fourier transform x(z) i— x(z) = X^^Lo x(n)zn

to the system equations (1.1), under the assumption of a zero initial condition

x(0) = 0 on the state vector, leads to the input-output relation

y(z) =Tv(z)u(z)

where T^(z) is the transfer function of the system

Tv(z) = D + zC(I -

zA)~1B.

When one starts from the point of view of operator-model theory, one is given

a contraction operator T on a Hilbert space H. Associated with T is a Schur-class

function TT(Z) G S(P,£*), the characteristic operator function for T, where V

and T* are the defect spaces associated with T. Given any Schur-class function

T, one can then build a model operator TV on a functional Hilbert space 7"^T so

that the characteristic function of T T is equal to T, and any completely nonunitary

contraction operator T with characteristic function equal to T is unitarily equivalent

to its model operator T^; in this way we see that the characteristic function T T is

a complete unitary invariant for completely nonunitary contraction operators T on

a Hilbert space Tt.

Note that the same Schur-class function S G S(£ ,£*) can serve as the scatter-

ing function for a Lax-Phillips system, as the transfer function for a conservative,

discrete-time, input-state-output linear system £(£/), a n d as the characteristic op-

erator function for a contraction operator T. Thus there is a natural association and

equivalence between Lax-Phillips scattering systems (5, conservative, discrete-time,

input-state-output linear systems £({/), and Hilbert-space contraction operators

T.

In the generalization which we discuss here, the unitary operator U in a Lax-

Phillips scattering system (5 is replaced by a d-tuple of isometries U = (Wi,... Md)

on a Hilbert space /C such that the block row \U\ ... Ud] : ®/5=i ^ ~~ * ^ ^s

unitary, i.e., (U\,... Md) provides an isometric representation of the Cuntz algebra

Od on JC (see e.g. the book of Davidson [Da96] for basic definitions). To set up a

scattering framework for this situation we hypothesize the existence of subspaces Q

and 5* such that Q and Q* are invariant for U\,..., Ud and U{,..., Wd respectively,

and U\g is a "row shift" and U*\g^ is the "backwards row shift" (precise definitions

are given in Chapter 3). The Fourier representation theory is now more complicated,

but it turns out that one can represent the smallest closed subspace Q containing Q

and reducing for each of the operators IA\,..., Ud as a certain functional model space

Cw for Cuntz-algebra representations, with Q corresponding to a certain analogue

Hw of a Hardy subspace. Similarly, the smallest closed subspace Q* containing 5*

and reducing for each U\,... ,Ud can be represented as a Cuntz-algebra functional

model space Cw*-, with Q* corresponding to Ji^ . Then the restricted projection

operator Pg U corresponds to a certain multiplication operator f \-+ S - f where

S now is a certain formal power series in two d-tuples of noncommuting variables