SPEAKER: Professor Cora Sadosky, Howard University

TITLE: Function Theory on the Polydisk and Multidimensional Systems:  Scattering Systems and State Space Realizations

ABSTRACT: There is a natural one-to-one correspondence between one-dimensional conservative (linear time-invariant) discrete-time input/state/output (i/s/o) systems and (orthogonal or causal) scattering systems with one evolution operator as introduced by Adamyan--Arov following Lax--Phillips. The scattering function of the scattering system coincides with the transfer function of the corresponding i/s/o system. Following the work of Agler on the Nevanlinna--Pick interpolation for the polydisk, Ball--Trent and Kalyuzhniy-Verbovetsky have studied $d$-dimensional conservative discrete-time i/s/o systems of Roesser and Fornasini-Marchesini type respectively and their transfer functions. Here we introduce, following Cotlar--Sadosky, the notion of a scattering system with $d$ commuting evolution operators. The scattering function of such a scattering system (assuming the system is orthogonal or causal) is a Schur class function on the $d$ dimensional polydisk. To a given a $d$-dimensional conservative i/s/o system always corresponds a scattering system with $d$ evolution operators, with its scattering function equal to the transfer function of the i/s/o system. However, for $d > 1$ we cannot in general recover the i/s/o system uniquely from the corresponding scattering system, nor is it obvious apriori that given a scattering system such a recovery is possible. Let us define the scattering subspace as the orthogonal complement to the incoming and the outgoing subspaces in the total space of the scattering system. Then there is a one-to-one correspondence between different (non unitarily equivalent) i/s/o systems corresponding to the same scattering system, and suitable decompositions of the scattering subspace with respect to the evolution operators. For $d=1$ a suitable decomposition trivially exists and is unique. For $d=2$ a suitable decomposition always exists, though this is by no means a trivial fact; it is not unique. For $d>2$ a suitable decomposition does not necessarily exist (and when it does, it is not unique). In the case of $d=2$ it is possible to construct explicitly two extremal decompositions of the scattering subspace so that any decomposition is contained (in a certain sense) between these two. These extremal decompositions should play an important role in the study of interpolation problems on the polydisk.

   Unlike in the one-dimensional case, the definition of the incoming, the outgoing, and the scattering subspaces for a scattering system with $d$ evolution operators depends on the choice of the ``cut'' between the past and the future, i.e., on the choice of the future cone. Here  it is important to consider  all of these cuts and not just one. Two cuts play a special role: the ``minimal cut,'' making the future as small as possible, and the ``balanced cut,'' giving the future and the past identical shapes.

This is a joint work with J. Ball and V. Vinnikov.

PLACE: McMahon Hall, Room 209

ORGANIZERS: A. Levin, M. Ostrovskii and P. Saworotnow, The Catholic University
of America, and  A. Vogt, Georgetown University

Tel: (202)319-5221, (202)319-5222.

Fax: (202)319-5231.

E-mail: levin@cua.edu, ostrovskii@cua.edu