Journal of Operator Theory

Volume 43, Issue 2, Spring 2000 pp. 427-460.

Multiparametric dissipative linear stationary dynamical scattering systems: discrete case

Authors: Dmitriy S. Kalyuzhniy
Author institution: Department of Higher Mathematics, Odessa State Academy of Civil Engineering and Architecture, Didrihson str. 4, 270029, Odessa, Ukraine

Summary: We propose the new generalization of linear stationary dynamical systems with discrete time $t\in{\bbb Z}$ to the case $t\in\nspace{Z}{N}$. The dynamics of such a system can be reproduced by means of its associated multiparametric Lax-Phillips semigroup. We define multiparametric dissipative and conservative scattering systems and interpret them in terms of operator colligations, of the associated semigroup, and of ``energy'' relations for system data. We prove the Agler's type theorem describing the class of holomorphic operator-valued functions on the polydisc $\nspace{D}{N}$ that are the transfer functions of multiparametric conservative scattering systems.

Keywords: Dissipative systems, multiparametric Lax-Phillips semigroup, generalized Schur class, conservative realizations

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