Linear systems, operator model theory and scattering: Multivariable generalizations. (English) Zbl 0973.93021
Ramm, A. G. (ed.) et al., Operator theory and its applications. Proceedings of the international conference, Winnipeg, Canada, October 7-11, 1998. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 25, 151-178 (2000).
The article represents a review and also some development of the ideas of linear systems theory, Lax-Phillips scattering theory and the Sz.-Nagy-Foias model theory based on the Sz.-Nagy deletion theorem for a contraction operator.
All these theories are united by a contraction, analytic, operator-valued function on the unit disk \(W(z)\) having a representation of the form \(W(z)= \Delta+ zC(I- zA)^{-1}B\).
The following extensions are discussed in the article:
1) replacement of the unit disk by the unit ball in \(C^d\),
2) the unit polydisk in \(C^d\),
3) a component of the nonreal points on an algebraic curve embedded in \(C^2\) with antihomomorphic involution \((\lambda_1, \lambda_2)\to (\overline\lambda^{-1}_1, \overline\lambda^{- 1}_2)\).
For the entire collection see [Zbl 0943.00054].
All these theories are united by a contraction, analytic, operator-valued function on the unit disk \(W(z)\) having a representation of the form \(W(z)= \Delta+ zC(I- zA)^{-1}B\).
The following extensions are discussed in the article:
1) replacement of the unit disk by the unit ball in \(C^d\),
2) the unit polydisk in \(C^d\),
3) a component of the nonreal points on an algebraic curve embedded in \(C^2\) with antihomomorphic involution \((\lambda_1, \lambda_2)\to (\overline\lambda^{-1}_1, \overline\lambda^{- 1}_2)\).
For the entire collection see [Zbl 0943.00054].
Reviewer: Andreiy Kondrat’yev (Pensacola)
MSC:
| 93C35 | Multivariable systems, multidimensional control systems |
| 93C05 | Linear systems in control theory |
| 47A48 | Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. |
| 93B28 | Operator-theoretic methods |
