On the Potapov Theory of Multiplicative Representations

Ginzburg, Yu. P.; Shevchuk, Liudmyla

doi:10.1007/978-3-0348-8532-4_3

Cited by 3 publications

(4 citation statements)

References 6 publications

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“…Taking into account (2.33) and passing to the limit in (2.40)as R --* or we obtain It follows from this inequality that ~-I(A) is the maximal matrix function satisfying condition (3.1) (see [7]). Therefore, by the Ginzburg theorem [8,9], the representation (3.12) holds. The proof is complete.…”

Section: A Multiplicative Representation Of ~(A)mentioning

confidence: 88%

Spectral theory of a class of canonical differential systems

Sakhnovich¹

2000

Funct Anal Its Appl

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Section: A Multiplicative Representation Of ~(A)mentioning

confidence: 88%

Spectral theory of a class of canonical differential systems

Sakhnovich¹

2000

Funct Anal Its Appl

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“…Later he together with his students and colleagues applied the theory of J-contractive matrix functions to different problems in analysis and passive electrical network theory, see [19], [20], [22], [15], and the references listed there. Potapov's results on matrix-valued functions were generalized to the operator-valued case by Ginzburg; see [17] and its references.…”

Section: It Follows From Theorem 84 That If D Is An Arbitrary Functimentioning

confidence: 97%

“…17) In Section 6 we discuss realizations of a given matrix-valued or operatorvalued function as the generalized i/o transfer function of a s/s system corresponding to an i/o decomposition W = Y U of the signal space. We begin by investigating how one can construct a s/s realization Σ ζ of a shifted generalized transfer function z → D(z + ζ) in terms of the originally given s/s system Σ.…”

Section: Introductionmentioning

confidence: 99%

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State/Signal Linear Time-Invariant Systems Theory, Part IV: Affine and Generalized Transfer Functions of Discrete Time Systems

Arov

Staffans

2007

Complex anal.oper. theory

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In the present article we continue to develop the linear discrete time-invariant state/signal systems theory that we have introduced and studied in three earlier articles (Parts I-III). The trajectories of a state/signal system Σ with a Hilbert state space X and a Hilbert or Kreȋn signal space W consists of a pair of sequences (x(·), w(·)) which after an admissible input/output decomposition W = Y U of the signal space can be obtained from the set of trajectories (x(·), u(·), y(·)) of a standard input/state/output system by taking w(·) = y(·) + u(·). In Part I we studied the families of all admissible decompositions of W for a given state/signal system Σ and the corresponding input/state/output representations and their transfer functions.Here we extend that theory to the the non-admissible case, and obtain generalized input/output transfer functions, as well as right and left affine transfer functions. As opposed to a standard transfer function, a generalized transfer function not need to be holomorphic at the origin. This makes it possible to realize a much larger class of transfer functions than what is possible by using the standard input/state/output theory. For example, every rational matrixvalued function (including those that have a pole at the origin) can be realized as the generalized transfer function of a state/signal system whose state space has a finite dimension equal to the McMillan degree of the given function. Likewise, every meromorphic J-contractive matrix-valued function can be realized as the generalized transfer function of a simple conservative state/signal system, or of a minimal passive state/signal system. Similar operator-valued results are also presented. As is well-known, a rational matrix-valued function with a pole at the origin can also be realized as the transfer function of a descriptor system, but this descriptor realization has the drawback that the dimension of its state space is bigger than the dimension of the state space of our s/s realization (i.e., bigger than the McMillan degree of the function). Our Damir Z. Arov thanksÅbo Akademi for its hospitality and the Academy of Finland and the Magnus Ehrnrooth Foundation for their financial support during his visits toÅbo in 2003-2006.458 D. Z. Arov and O. J. Staffans Comp.an.op.th.s/s realizations also differ significantly from other known realizations which depend on the choice of an auxiliary point in the complex plane where the given function is holomorphic. (2000). Primary 47A48, 93A05; Secondary 94C05. Mathematics Subject Classification

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On the theory of matrix-valued functions belonging to the Smirnov class

Katsnelson

Kirstein

1997

Topics in Interpolation Theory

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Abstract. A theory of matrix-valued functions from the matricial Smirnov class 1J1;t (]]JJ) is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman's theorem are presented for matrix-valued functions from the Smirnov class 1J1;t(]]JJ). We also consider a family F>.. = F -AI of functions belonging to the matricial Smirnov class which is indexed by a complex parameter A. We show that with the exception of a "very small" set of such A the corresponding inner factor in the inner-outer factorization of the function F>.. is a Blaschke-Potapov product.The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class. Notationse -the complex plane. '1:= {t E e : It I = I} -the unit circle. IIJ):= {z E e : Izl < I} -the unit disc.~1l' -the a-algebra of Borel subsets of'l. m -normalized Lebesgue measure on the measurable space ('I, ~1l').en -the n-dimensional complex space equipped with the usual Euclidean In -the n X n unit matrix.

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On the Potapov Theory of Multiplicative Representations

Cited by 3 publications

References 6 publications

Spectral theory of a class of canonical differential systems

Spectral theory of a class of canonical differential systems

State/Signal Linear Time-Invariant Systems Theory, Part IV: Affine and Generalized Transfer Functions of Discrete Time Systems

On the theory of matrix-valued functions belonging to the Smirnov class

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