Linear Fractional Transformations of Nevanlinna Functions Associated with a Nonnegative Operator

Behrndt, Jussi; Hassi, Seppo; Snoo, Henk de; Wietsma, Rudi; Winkler, Henrik

doi:10.1007/s11785-011-0197-3

Cited by 6 publications

(2 citation statements)

References 15 publications

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“…Other such characterizations (q 2 N Ä 1 such that zq.z/ 2 N Ä 2 and others) have been studied and characterized in terms of the representing relations; see [42] and references therein, and also [6].…”

Section: Some Special Subclassesmentioning

confidence: 99%

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: Some Special Subclassesmentioning

confidence: 99%

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Luger

2014

Operator Theory

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“…Denote by S M the set of Nevanlinna functions that satisfy the equivalent conditions in Lemma 2.9. The set S M appears in [8] as the union of Nevanlinna functions of type I, II, III and V ′ , see [8,Definition 2.4] and also [8,Corollary 5.6].…”

Section: 2mentioning

confidence: 99%

Partially fundamentally reducible operators in Krein spaces

Ćurgus¹,

Derkach²

2014

Preprint

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A self-adjoint operator A in a Kreȋn space (K, [ • , • ]) is called partially fundamentally reducible if there exist a fundamental decomposition K = K + [ +]K − (which does not reduce A) and densely defined symmetric operators S + and S − in the Hilbert spaces (K + , [ • , • ]) and (K − , −[ • , • ]), respectively, such that each S + and S − has defect numbers (1, 1) and the operator A is a self-adjoint extension of S = S + ⊕ (−S − ) in the Kreȋn space (K, [ • , • ]). The operator A is interpreted as a coupling of operators S + and −S − relative to some boundary triples C, Γ + 0 , Γ + 1 and C, Γ − 0 , Γ − 1 . Sufficient conditions for a nonnegative partially fundamentally reducible operator A to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions m + and m − of S + and S − relative to the boundary triples C, Γ + 0 , Γ + 1 and C, Γ − 0 , Γ − 1 . Moreover, it is shown that under some asymptotic assumptions on m + and m − all positive self-adjoint extensions of the operator S are similar to self-adjoint operators in a Hilbert space.

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Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Luger

2015

Operator Theory

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Linear Fractional Transformations of Nevanlinna Functions Associated with a Nonnegative Operator

Cited by 6 publications

References 15 publications

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Partially fundamentally reducible operators in Krein spaces

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

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