On realization of the Kreĭn–Langer class <i>N<sub>κ</sub></i> of matrix‐valued functions in Pontryagin spaces

Arlinskii̇̆, Yury; Belyi, Sergey; Derkach, Vladimir; Tsekanovskiı̆, Eduard

doi:10.1002/mana.200510686

Cited by 6 publications

(4 citation statements)

References 28 publications

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“…One such example is that of linear fractional transformations of the transfer function of a linear stationary conservative dynamic system (also called the Brodski-Livsic rigged operator colligation); see, e.g., [4].…”

Section: Other Representationsmentioning

confidence: 99%

“…Note that this is not a canonical factorization as in (4). The statement can be proven via inspection of the generalized zeros and approximation arguments; see [33].…”

Section: A Function-theoretic Characterizationmentioning

confidence: 99%

“…and hence it is a pole and a zero for Q 4 . Note that in the definition of a generalized zero, it was assumed that Q is invertible.…”

Section: Spectrum Versus Singularitiesmentioning

confidence: 99%

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Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Luger

2014

Operator Theory

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This article gives an introduction and short overview on generalized Nevanlinna functions, with special focus on asymptotic behavior and its relation to the operator representation. Introduction: Classical Nevanlinna FunctionsGeneralized Nevanlinna functions (scalar, matrix-or operator-valued) are functions that are meromorphic in CnR satisfying certain symmetry and sign conditions. They appeared first in connection with self-adjoint operators and relations in Pontryagin spaces (in [46] for scalar and [47] for matrix-valued functions) and have become an important tool in extension theory, e.g., for differential operators in connection with eigenvalue dependent boundary conditions. This survey is started with a short review of old and well-known facts about classical Nevanlinna functions (also known as Herglotz-, Pick-, or R-functions), which build a basis for the indefinite generalizations.Recall that a function q that maps the upper half plane C C holomorphically into C C [R is called a Nevanlinna function, q 2 N 0 .C/. These functions, which appear in many applications, e.g., as Titchmarsh-Weyl coefficients in Sturm-Liouville problems, are very well studied objects. In particular, such a function admits an integral representation ([37,58,59], presented in the following form by Cauer [16]; see also [1]):where a 2 R, b 0, and a positive Borel measure with R R d .t/ 1Ct 2 < 1. More abstractly, for every such function there exists a Hilbert space K; OE ; , a self-adjoint linear relation (i.e., a multi-valued operator) A in K and an element v in K such that with some z 0 2 %.A/ the function q can be written as q.z/ D q.z 0 / C .z z 0 / I C .z z 0 /.A z/ 1 v; v K :E-mail: luger@math.su.se Page 1 of 24Operator Theory DOI 10.1007/978-3-0348-0692-3_35-1 © Springer Basel 2015 In particular, for b D 0 one can choose K D L 2 , where is the measure in the integral representation (1), then A is the operator of multiplication by the independent variable. If b ¤ 0 then A is not an operator, but a relation with one-dimensional multi-valued part (see, e.g., [32] for the theory of linear relations and for the details of this representation, e.g., [53]).The limit behavior of q at the real line can be deduced directly from the above representations. In particular, lim z O !˛.˛ z/q.z/, where lim z O !˛d enotes the non-tangential limit to˛2 R, always exists and is zero or positive. Here the second case is equivalent to˛being an eigenvalue of the minimal representing relation A in (2), or, equivalently, f˛g ¤ 0.In several examples, however, e.g., in connection with singular Sturm-Liouville problems (see, e.g., [31,50,52]), there appear functions that are not as described above, but belong to a so-called generalized Nevanlinna class. Such functions may have non-real poles and the limit lim z O !˛.˛ z/q.z/ does not necessarily exist. However, this exceptional behavior can appear at finitely many points only, which relies on the fact that these functions admit operator representations of the form (2) but in Pontryagin spaces....

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Section: Other Representationsmentioning

confidence: 99%

“…Note that this is not a canonical factorization as in (4). The statement can be proven via inspection of the generalized zeros and approximation arguments; see [33].…”

Section: A Function-theoretic Characterizationmentioning

confidence: 99%

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Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Luger

2014

Operator Theory

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“…Finally we mention the works [17,18,38] to stress the interest of positive and generalized positive functions in linear system theory and operator theory.…”

Section: Introductionmentioning

confidence: 99%

Realizations of Slice Hyperholomorphic Generalized Contractive and Positive Functions

et al. 2014

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Introduction 1 1.1. Schur analysis 2 1.2. Negative squares 2 1.3. The slice hyperholomorphic case 3 2. Preliminaries 4 2.1. Negative squares and kernels 4 2.2. Slice hyperholomorphic functions 6 3. Slice hyperholomorphic operator-valued functions 8 3.1. Realizations 13 4. The Hardy space of the half-space H + 14 5. The Schauder-Tychonoff fixed point theorem 21 5.1. The Schauder-Tychonoff fixed point theorem 21 5.2. An invariant subspace theorem 23 6. The spaces P(S) 24 7. Realization for elements in S κ (J 1 , J 2 ) 26 8. The space L(Φ) and realizations for generalized positive functions 33 8.1. The indefinite case 33 8.2. The positive case 40 References 42

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Truncated moment problems in the class of generalized Nevanlinna functions

Derkach

Hassi

Snoo

2012

Mathematische Nachrichten

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Truncated moment and related interpolation problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established and complete parametrizations of solutions are given. The framework used in the paper allows the treatment of even and odd order problems in a parallel manner. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new effects in the indefinite case is that the degenerate moment problem may have infinitely many solutions. However, with a careful application of an indefinite analogue of a step-by-step Schur algorithm a complete description of the set of solutions will be obtained.

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On realization of the Kreĭn–Langer class N_κ of matrix‐valued functions in Pontryagin spaces

Cited by 6 publications

References 28 publications

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Realizations of Slice Hyperholomorphic Generalized Contractive and Positive Functions

Truncated moment problems in the class of generalized Nevanlinna functions

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On realization of the Kreĭn–Langer class Nκ of matrix‐valued functions in Pontryagin spaces

Cited by 6 publications

References 28 publications

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Realizations of Slice Hyperholomorphic Generalized Contractive and Positive Functions

Truncated moment problems in the class of generalized Nevanlinna functions

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On realization of the Kreĭn–Langer class N_κ of matrix‐valued functions in Pontryagin spaces