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E-raamat: State Space Method: Generalizations and Applications

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State Space Method: Generalizations and ApplicationsSuurem pilt
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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.
Editorial Introduction ix
Discrete Analogs of Canonical Systems with Pseudo-exponential Potential, Definitions and Formulas for the Spectral Matrix Functions
1(48)
D. Alpay
I. Gohberg
Introduction
2(2)
Review of the continuous case
4(15)
The asymptotic equivalence matrix function
4(4)
The other characteristic spectral functions
8(6)
The continuous orthogonal polynomials
14(2)
Perturbations
16(3)
The discrete case
19(20)
First-order discrete system
19(3)
The asymptotic equivalence matrix function
22(5)
The reflection coefficient function and the Schur algorithm
27(2)
The scattering function
29(2)
The Weyl function and the spectral function
31(2)
The orthogonal polynomials
33(4)
The spectral function and isometries
37(2)
Two-sided systems and an example
39(10)
Two-sided discrete first-order systems
39(2)
An illustrative example
41(3)
References
44(5)
Matrix-J-unitary Non-commutative Rational Formal Power Series
49(66)
D. Alpay
D.S. Kalyuzhnyi-Verbovetzkii
Introduction
51(3)
Preliminaries
54(6)
More on observability, controllability, and minimality in the non-commutative setting
60(7)
Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the line case
67(10)
Minimal Givone--Roesser realizations and the Lyapunov equation
68(4)
The associated structured Hermitian matrix
72(2)
Minimal matrix-J-unitary factorizations
74(1)
Matrix-unitary rational formal power series
75(2)
Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the circle case
77(10)
Minimal Givone--Roesser realizations and the Stein equation
77(6)
The associated structured Hermitian matrix
83(1)
Minimal matrix-J-unitary factorizations
84(1)
Matrix-unitary rational formal power series
85(2)
Matrix-J-inner rational formal power series
87(9)
A multivariable non-commutative analogue of the half-plane case
87(4)
A multivariable non-commutative analogue of the disk case
91(5)
Matrix-selfadjoint rational formal power series
96(6)
A multivariable non-commutative analogue of the line case
96(4)
A multivariable non-commutative analogue of the circle case
100(2)
Finite-dimensional de Branges--Rovnyak spaces and backward shift realizations: The multivariable non-commutative setting
102(13)
Non-commutative formal reproducing kernel Pontryagin spaces
102(4)
Minimal realizations in non-commutative de Branges--Rovnyak spaces
106(4)
Examples
110(1)
References
111(4)
State/Signal Linear Time-Invariant Systems Theory, Part I: Discrete Time Systems
115(64)
D.Z. Arov
O.J. Staffans
Introduction
116(4)
State/signal nodes and trajectories
120(3)
The driving variable representation
123(5)
The output nulling representation
128(4)
The input/state/output representation
132(6)
Transfer functions
138(8)
Signal behaviors, external equivalence, and similarity
146(7)
Dilations of state/signal systems
153(14)
Stability
167(9)
Appendix
176(3)
Acknowlegment
176(1)
References
176(3)
Conservative Structured Noncommutative Multidimensional Linear Systems
179(46)
J.A. Ball
G. Groenewald
T. Malakorn
Introduction
180(3)
Structured noncommutative multidimensional linear systems: basic definitions and properties
183(8)
Adjoint systems
191(2)
Dissipative and conservative structured multidimensional linear systems
193(6)
Conservative SNMLS-realization of formal power series in the class SAG (U, Y)
199(26)
References
220(5)
The Bezout Integral Operator: Main Property and Underlying Abstract Scheme
225(43)
I. Gohberg
I. Haimovici
M.A. Kaashoek
L. Lerer
Introduction
226(2)
Spectral theory of entire matrix functions
228(13)
A review of the spectral data of an analytic matrix function
229(3)
Eigenvalues and Jordan chains in terms of realizations
232(2)
Common eigenvalues and common Jordan chains in terms of realizations
234(3)
Common spectral data of entire matrix functions
237(4)
The null space of the Bezout integral operator
241(15)
Preliminaries on convolution integral operators
242(2)
Co-realizations for the functions A, B, C, D
244(4)
Quasi commutativity in operator form
248(3)
Intertwining properties
251(3)
Proof of the first main theorem on the Bezout integral operator
254(2)
A general scheme for defining Bezout operators
256(12)
A preliminary proposition
257(3)
Definition of an abstract Bezout operator
260(2)
The Haimovici-Lerer scheme for defining an abstract Bezout operator
262(2)
The Bezout integral operator revisited
264(2)
The null space of the Bezout integral operator
266(2)
References
268


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