The State Space Method

The state space method developed in the last decades allows us to study the theory of linear systems by using tools from the theory of linear operators; conversely, it had a strong influence on operator theory introducing new questions and topics.

The present volume contains a collection of essays representing some of the recent advances in the state space method. Methods covered include noncommutative systems theory, new aspects of the theory of discrete systems, discrete analogs of canonical systems, and new applications to the theory of Bezoutiants and convolution equations.

The articles in the volume will be of interest to pure and applied mathematicians, electrical engineers and theoretical physicists.

Résumé
Thisvolumeofthe Operator Theory: Advances and Applications series (OTAA) isthe ?rst volume of a new subseries. This subseries is dedicated to connections between the theory of linear operators and the mathematical theory of linear systems and is named Linear Operators and Linear Systems (LOLS).Asthe- isting subseries Advances in Partial Di?erential Equations (ADPE), the new s- series will continue the traditions of the OTAA series and keep the high quality of the volumes. The editors of the new subseries are: Daniel Alpay (BeerSheva, - rael), Joseph Ball (Blacksburg, Virginia, USA) and Andr´ ´ e Ran (Amsterdam, The Netherlands). In the last 2530 years, Mathematical System Theory developed in an ess- tial way. A large part of this development was connected with the use of the state space method. Let us mention for instance the theory of H control. The state ? space method allowed to introduce in system theory the modern tools of matrix and operator theory. On the other hand the state space approach had an imp- tant impact on Algebra, Analysis and Operator Theory. In particular it allowed to solve explicitly some problems from interpolation theory, theory of convolution equations, inverse problems for canonical di?erential equations and their discrete analogs. All these directions are planned to be present in the subseries LOLS. The editors and the publisher are inviting authors to submit their manuscripts for publication in this subseries.

Contenu
Discrete Analogs of Canonical Systems with Pseudo-exponential Potential. Definitions and Formulas for the Spectral Matrix Functions.- Matrix-J-unitary Non-commutative Rational Formal Power Series.- State/Signal Linear Time-Invariant Systems Theory, Part I: Discrete Time Systems.- Conservative Structured Noncommutative Multidimensional Linear Systems.- The Bezout Integral Operator: Main Property and Underlying Abstract Scheme.