1. INTRODUCTION

5

(5) By the Halmos dilation process, a completely nonunitary row contraction

T = (Ti,... , Td) uniquely determines a "strict", closely-connected Cuntz

unitary colligation U (1.2) (with (Tu ..., Td) = (AJ,..., -AJ)). Hence a

complete set of unitary invariants is the pair (T, L) where T is the charac-

teristic function for T (i.e., the characteristic function for the colligation

U) and L is a choice of Haplitz extension of the Cuntz-Toeplitz defect op-

erator I — (MT)*MT. In fact, the row contraction T is unitarily equivalent

to its Sz.-Nagy-Foia§ model row contraction T(

T L

) constructed from its

characteristic pair (T, L). This removes the "completely noncoisometric"

restriction in the result of Popescu cited above.

The paper is organized as follows. After the present Introduction, in Chapter 2

we recall the formalism from [BaV04] giving an analogue of L2-spaces which serves

as a model for row unitary operators. We then study in some detail the analogues

of the Hardy space Hw

a n

d H^ for this setting, as well as the analogues of L°° and

H°°, namely, the space of intertwining maps LT between two row-unitary model

spaces Cw and Cw* •

a n (

i the subclass of such maps ("analytic intertwining opera-

tors") which preserve the associated Hardy spaces (My: Hw — Hw*)- Of particu-

lar interest is to understand how to extend an intertwining map MT • Hw ~^ Hw*

defined only on a Hardy space to a full intertwining map LT : Cw — &w*- The

contractive, analytic intertwining operators then form an interesting noncommu-

tative analogue of the "Schur class" which has been receiving much attention of

late from a number of points of view (see e.g. [BaOO]). Preliminary to this anal-

ysis is understanding how to extend a (noncommutative) Hardy space Hw+ to a

full noncommutative Lebesgue space Cw, or equivalently, how to extend a given

noncommutative Toeplitz operator

W+

to a full Haplitz operator W. An equiv-

alent operator-theoretic formulation is how to extend a given row isometry to a

row unitary; unlike as in the classical case, this extension generally is not unique.

Our analysis of these issues is via a noncommutative analogue of the Szego factor-

ization theory for a nonnegative Toeplitz operator; here there is overlap with the

seminal wTork of Popescu (see [PoOlb]), who also gives noncommutative analogues

of the connections with prediction theory and entropy optimization, and with the

work of Adams, Froelich, McGuire and Paulsen (see [AFMP94]), who discuss such

factorization results in a commutative but non-Toeplitz setting.

In Chapter 3, we axiomatize the notion of a "Cuntz Lax-Phillips scattering sys-

tem" , define the scattering function for such an object and the associated functional

models (of Pavlov, de Branges-Rovnyak and Sz.-Nagy-Foia§ type) for any minimal

Cuntz scattering system. Chapter 4 introduces the multidimensional system E and

unitary colligation corresponding to a Cuntz Lax-Phillips scattering system, along

with the associated transfer function characterizing input-output properties and the

mappings back and forth between Cuntz scattering systems and Cuntz unitary colli-

gations. The final Chapter 5 presents the results on unitary classification of Cuntz

unitary colligations, or equivalently, of completely nonunitary row contractions.

Here we also explain how the curvature invariant curv(T) for a row contraction

T = (Ti,...,Td) with finite-rank defect operator D? introduced in [KrOl] and

[PoOla] can be computed directly in terms of the characteristic function TT(Z) of

T.