TY - JOUR

T1 - Scattering systems with several evolutions and multidimensional input/state/output systems

AU - Ball, Joseph A.

AU - Sadosky, Cora

AU - Vinnikov, Victor

N1 - Funding Information:
J. Ball was partially supported by US National Science Foundation under Grant Number DMS-9987636. C. Sadosky was partially supported by Department of Energy grant DE-FG02-ER25341. All the authors were supported in part by US-Israel Binational Science Foundation grants 9900252 and 2002414.

PY - 2005/7/1

Y1 - 2005/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=23644432815&partnerID=8YFLogxK

U2 - 10.1007/s00020-005-1351-y

DO - 10.1007/s00020-005-1351-y

M3 - Article

AN - SCOPUS:23644432815

VL - 52

SP - 323

EP - 393

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 3

ER -