Colligations in Pontryagin Spaces with a Symmetric Characteristic Function

Alpay, Daniel; Azizov, T. Ya.; Dijksma, Aad; Rovnyak, James

doi:10.1007/978-3-0348-8181-4_7

Cited by 8 publications

(9 citation statements)

References 3 publications

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“…Suppose next the simple conservative realization of the symmetric function θ ∈ S κ1 (U) in Theorem 3.8 is chosen to be the canonical unitary realization. Then it can be derived from [3,Theorem 3.6] In addition, if also θ ∈ U κ1 (U), it has been shown in the proof of Theorem 3.8 that all co-isometric observable or isometric controllable realizations of θ are minimal conservative. Therefore it can be assumed that Σ 1 in Theorem 3.8 is the canonical co-isometric realization.…”

Section: )mentioning

confidence: 99%

“…If U and Y are Pontryagin spaces with the same negative index, one encounters operator colligations, or systems, such that all the spaces are indefinite. Theory of canonical isometric, co-isometric and conservative systems in that case is considered, for instance, in [2,3,20,23], along with the other properties of the generalized Schur functions. Especially, symmetric generalized Schur functions, with a little bit more general definition than in this paper, were studied in [3].…”

Section: Introductionmentioning

confidence: 99%

“…That is, the negative index of the Z, which roughly speaking tells that how much Z behaves like a positive operator with respect to the indefinite inner product in question, can be used to determine whether θ is also a generalized Nevanlinna function. If, in addition, the boundary values of θ on the unit disc T are unitary, then Z is also unitary and can be represented in an explicit form by using the canonical realizations from [2,3].…”

Section: Introductionmentioning

confidence: 99%

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Generalized Schur–Nevanlinna functions and their realizations

Lilleberg

2020

Integr. Equ. Oper. Theory

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Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which allows one to generalize the notion of an inner function to Pontryagin space operator valued setting. Transfer functions of self-adjoint systems such that their state spaces are Pontryagin spaces, are generalized Nevanlinna functions, and symmetric generalized Schur functions can be realized as transfer functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion when a symmetric generalized Schur function is also a generalized Nevanlinna function is given. The criterion involves the negative index of the weak similarity mapping between an optimal minimal realization and its dual. In the special case corresponding to the generalization of an inner function, a concrete model for the weak similarity mapping can be obtained by using the canonical realizations.

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Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Schur–Nevanlinna functions and their realizations

Lilleberg

2020

Integr. Equ. Oper. Theory

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“…The case where the state space is a Pontryagin space while incoming and outgoing spaces are still Hilbert spaces, unitary systems were studied, for instance, by Dijksma et al [22,23], and passive systems by Saprikin [28], Saprikin and Arov [12], Saprikin et al [11] and by the author in [27]. The case where all the spaces are Pontryagin spaces, theory of isometric, co-isometric and conservative systems is considered, for instance, in [1,2,24].…”

Section: Introductionmentioning

confidence: 99%

Minimal Passive Realizations of Generalized Schur Functions in Pontryagin Spaces

Lilleberg

2020

Complex Anal. Oper. Theory

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Passive discrete-time systems in Pontryagin space setting are investigated. In this case the transfer functions of passive systems, or characteristic functions of contractive operator colligations, are generalized Schur functions. The existence of optimal and * -optimal minimal realizations for generalized Schur functions are proved. By using those realizations, a new definition, which covers the case of generalized Schur functions, is given for defects functions. A criterion due to D.Z. Arov and M.A. Nudelman, when all minimal passive realizations of the same Schur function are unitarily similar, is generalized to the class of generalized Schur functions. The approach used here is new; it relies completely on the theory of passive systems.

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“…see, for instance, [21,Theorem 3.5]. The theory of reproducing kernel Pontryagin spaces of the form P(s) can be found in [19], see also [12].…”

Section: Additional Remarks and Referencesmentioning

confidence: 99%

The Transformation of Issai Schur and Related Topics in an Indefinite Setting

Alpay

Dijksma

Langer

System Theory, the Schur Algorithm and Multidimensional Analysis

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We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation problems and problems of factorization of rational J-unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces and linear relations in these spaces.

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Colligations in Pontryagin Spaces with a Symmetric Characteristic Function

Cited by 8 publications

References 3 publications

Generalized Schur–Nevanlinna functions and their realizations

Generalized Schur–Nevanlinna functions and their realizations

Minimal Passive Realizations of Generalized Schur Functions in Pontryagin Spaces

The Transformation of Issai Schur and Related Topics in an Indefinite Setting

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