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.
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