CHAPTER 1
Introduction
It is well known that unitary linear system theory, scattering theory, and oper-
ator model theory are all closely related; for good explanations of various facets of
these theories and of their connections with each other, we refer to [Bro71, Li73,
NaF70, BraR66, BaC91, ADRS97, LaP67, AA95, NiV89, NiV98, BaTOO]
and the survey article [BaOO]. Indeed, there is essentially a one-to-one correspon-
dence between each pair of these objects (once some natural minimality and nor-
malization requirements are imposed). Each theory produces a contractive operator
function W(z) on the unit disk (called the transfer function of the linear system,
the scattering function of the scattering system, and the characteristic function of
the completely nonunitary (c.n.u.) contraction operator, respectively) from which
one can recover (up to unitary equivalence) the original object (unitary system,
scattering system, c.n.u. contraction operator, respectively). One can produce the
one-to-one correspondence between each pair of objects by way of this operator
function or via a more direct mapping, the details of which depend on the pair
under consideration.
A Lax-Phillips scattering system (as generalized by Adamjan and Arov) amounts
to a Hilbert space /C together with a unitary operator hi on /C and two subspaces
Q (the outgoing space) and Q* (the incoming space), invariant for hi and hi* re-
spectively, such that hi\g is a unilateral shift operator with wandering subspace
£ = Q GhiQ and U*\g^ is a unilateral shift operator with wandering subspace U*£*
(where £* = UG*OG*)- One can then use a natural Fourier representation to repre-
sent Q = closed span
n= 1 2
hi*nQ as I/2(T,£) (the L2-space of ^-valued functions
on the unit circle T) with hi\g represented as multiplication by the coordinate func-
tion z, and with the subspace Q dQ represented as the Hardy subspace H2(T,£).
Similarly, one can represent Q* closed spann=12)...
UnQ*
as
L2(T,
£*) with U\g
represented as multiplication by z on L2(T,£*) and with the subspace £?* C £*
represented as i72(T, £*)±. The restricted orthogonal projection Pg \g then corre-
sponds to a multiplication operator f(z) i-+ S(z)f(z) where S(z) G S(£,£*) (the
Schur class of holomorphic functions on the unit disk with values equal to con-
tractive operators between £ and £*) is the scattering function for the system.
Conversely, given a Schur-class function S G «S(£,£*), one can construct a model
scattering system (3s which has S has its scattering function, and any minimal
scattering system with scattering function equal to S is unitarily equivalent to its
model scattering system ©5. In this way one sees that the scattering function S is
a complete unitary invariant for minimal scattering systems.
A unitary colligation amounts to a block unitary operator U of the form
'A
B
C D £
-
1
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