Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Luger, Annemarie

doi:10.1007/978-3-0348-0667-1_35

Cited by 3 publications

(4 citation statements)

References 40 publications

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“…Moreover, it holds that q(ζ) = q(ζ) and q(C + ) ⊂ C + , where C + is the upper half plane. The details of this example and all subsequent proofs in this section can be found in the survey [20].…”

Section: Realizations On Hilbert Spacesmentioning

confidence: 99%

“…When h is rational, then q belongs to the well studied class of Generalized Nevanlinna functions. For this function class minimal realizations have been constructed in the past, see [20] for a survey.…”

Section: The Extended Nevanlinna Classmentioning

confidence: 99%

“…The following proposition collects properties of the defect function [20]: Proposition 4.3. Let q : D ⊂ C → C be analytic, (A, v) a realization and ϕ the defect function.…”

Section: A General Realization Theorymentioning

confidence: 99%

“…For example, on Pontryagin spaces matrix valued Generalized Nevanlinna functions are considered in e.g. [20] .…”

Section: Real Quasi-herglotz Functionsmentioning

confidence: 99%

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Realizations of Meromorphic Functions of Bounded Type

Emmel,

Luger

2023

Operator Theory: Advances and Applications

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In this article, we construct operator models for meromorphic functions of bounded type on Krein spaces. This construction is based on certain reproducing kernel Hilbert spaces which are closely related to model spaces. Specifically, we show that each function of bounded type corresponds naturally to a pair of such spaces, extending Helson's representation theorem. This correspondence enables an explicit construction of our model, where the Krein space is a suitable sum of these identified spaces. Additionally, we establish that the representing self-adjoint relations possess a relatively simple structure, proving to be partially fundamentally reducible. Conversely, we show that realizations involving such relations correspond to functions of bounded type.

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Section: Realizations On Hilbert Spacesmentioning

confidence: 99%

Section: The Extended Nevanlinna Classmentioning

confidence: 99%

“…The following proposition collects properties of the defect function [20]: Proposition 4.3. Let q : D ⊂ C → C be analytic, (A, v) a realization and ϕ the defect function.…”

Section: A General Realization Theorymentioning

confidence: 99%

“…For example, on Pontryagin spaces matrix valued Generalized Nevanlinna functions are considered in e.g. [20] .…”

Section: Real Quasi-herglotz Functionsmentioning

confidence: 99%

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Realizations of Meromorphic Functions of Bounded Type

Emmel,

Luger

2023

Operator Theory: Advances and Applications

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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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Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

Borogovac

2022

Ukr. Mat. Zhurn.

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UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function Q represented by a self-adjoint linear relation A in a Pontryagin space is decomposed by means of the reducing subspaces of A . The sum of two functions Q i ∈ N κ i ( ℋ ) , i = 1,2 , minimally represented by the triplets ( 𝒦 i , A i , Γ i ) is also studied. For this purpose, we create a model ( 𝒦 ˜ , A ˜ , Γ ˜ ) to represent Q : = Q 1 + Q 2 in terms of ( 𝒦 i , A i , Γ i ) . By using this model, necessary and sufficient conditions for κ = κ 1 + κ 2 are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation A affect the reducing subspaces of A and the decomposition of the corresponding function Q .

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Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Cited by 3 publications

References 40 publications

Realizations of Meromorphic Functions of Bounded Type

Realizations of Meromorphic Functions of Bounded Type

Generalized Schur–Nevanlinna functions and their realizations

Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

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