4

1. INTRODUCTION

H, we consider a d-tuple T = ( T i , . . . , T ^ ) of operators on a Hilbert space for

which the block row operator [Ti .. . T/| : ®/?=1 H —- TL is a contraction (a

row contraction); one obtains a Sz.-Nagy-Foia§-type operator model by analyzing

the geometry of the space of its minimal row-isometric dilation (see [Pr82, Bu84 ,

Po89a , Po89b]); in particular Popescu [Po89b] has shown that the characteristic

function for a row contraction is a complete unitary invariant in case the row con-

traction is "completely noncoisometric". In addition, we now have a Wold decom-

position for a row isometry [Pr84, Po95] together with a Beurling-Lax Theorem

[Po89b, DaP99 ] and Sz.-Nagy-Foia§ Commutant Lifting Theorem for this setting

[Po89c, Po95], a von Neumann inequality [Po99], a noncommutative analogue

of the C*-algebra of analytic Toeplitz operators [DaP98a], and a noncommuta-

tive analogue of Nevanlinna-Pick interpolation [DaP98b, Po98 , APOO, CSK02].

When one symmetrizes the underlying Fock space, one arrives at a reproducing

kernel Hilbert space H(kd) of analytic functions over the unit ball in Cd and its

associated space of bounded multipliers for which many parallel results hold (see

[Dr78, P o 9 1 , Arv98 , AMOO, B a T V O l , McTOO]). Moreover, we now have a

fairly complete understanding of a certain partial invariant derived from the char-

acteristic function, namely the "curvature invariant" (see [KrOl, PoOla] for the

general setting and [ArvOO, GRS02] for the commutative setting where the idea

actually originated).

The main focus of the paper is to lay out the connections between scatter-

ing, conservative linear systems and operator model theory for contractions in the

Cuntz-algebra setting. The following points summarize our main results:

(1) A Cuntz scattering system 6 , if minimal is determined up to unitary

equivalence by its scattering operator S&, and is unitarily equivalent to

any of three model Cuntz scattering systems built from its scattering

function.

(2) A Cuntz scattering system 6 determines a unique Cuntz unitary colli-

gation U — U(&); the transfer function T^jj) f°r the associated Cuntz

conservative linear system £(£/) is a certain restriction of the scattering

function SQ of the Cuntz scattering system.

(3) Conversely, a Cuntz unitary colligation U, together with a certain addi-

tional invariant W* (a ushift-like Cuntz weight" on output signals of the

system Ti(U)) uniquely determines a Cuntz scattering system 6(17, W*). If

the Cuntz unitary colligation U has the form U = U(&) for a Cuntz scat-

tering system 6 , then there is a particular choice of weight W* = W*(&)

determined by 6 so that we recover 6 as 6 = 6(Z7, W*). Moreover, the

outgoing shift Cuntz weight W and the scattering operator S for &(U, W*)

associated with 6(C/, W*) can be given explicitly in terms ofU and W*.

(4) A Cuntz unitary colligation uniquely determines its characteristic func-

tion T(z) — ^2ve:F Tvzv (a formal power series in the noncommutative, d-

variable Schur-class S

n c

,d(£,£*)— s e e (2.4.1) for the precise definition) as

well as a particular choice of "Haplitz" extension L of the Cuntz-Toeplitz

defect operator I — (Mr)*Mr- Moreover, the colligation can be recov-

ered from (T, L) up to unitary equivalence from a Sz.-Nagy-Foias model

constructed from (T,L); hence (T,L) forms a complete set of unitary

invariants for a closely-connected Cuntz-unitary colligation.