Scattering systems with several evolutions and multidimensional input/state/output systems

Joseph A. Ball, Cora Sadosky, Victor Vinnikov

Research output: Contribution to journalArticlepeer-review

40 Scopus citations

Abstract

The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô's dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis.

Original languageEnglish
Pages (from-to)323-393
Number of pages71
JournalIntegral Equations and Operator Theory
Volume52
Issue number3
DOIs
StatePublished - 1 Jul 2005

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