Multivariable linear systems, scattering, unitary dilations and operator model theory for row contractions
by
Joseph A. Ball
Virginia Tech
Coauthors: Victor Vinnikov (Weizmann Institute of Science, Rehovot, ISRAEL)

It is well known that unitary linear system theory, Lax-Phillips scattering theory, and Sz.-Nagy-Foias model theory are all closely related. Indeed, 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 abstract scattering system, or the characteristic operator function of the given contraction operator, respectively) from which, under natural minimality assumptions, one can recover the original object (i.e., unitary system, abstract scattering system or completely nonunitary contraction operator, respectively) up to unitary equivalence. The talk will discuss a generalization of these ideas to a multivariable setting, where a single contracton operator T is replaced by a d-tule (T₁, ..., T_d) of operators on a Hilbert space H for which the block row operator [ T₁ ... T_d ] is contractive. While the operator model theory aspects of this setting have been noted in the literature, we here integrate into the picture the previously neglected system-theoretic and scattering aspects. The transfer (or scattering or characteristic) function in this setting belongs to a certain noncommutative generalization of the multiplier space for the reproducing kernel space H(k_d) over the unit ball B^d in d dimensional unitary space with reproducing kernel equal to k_d(z, w) = 1/(1 - <z, w >).

Date received: March 22, 1999