Partially Fundamentally Reducible Operators in Kreĭn Spaces

Ćurgus, Branko; Derkach, Vladimir

doi:10.1007/s00020-014-2204-3

Cited by 4 publications

(9 citation statements)

References 40 publications

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“…In Proposition 5.1, it is shown that the operator A is definitizable over a vicinity of ∞ and conditions (4.18) for ∞ ∈ c s (A) are formulated in terms of the m-coefficients for the operators B ± . In the case w ≡ 1, the conditions (4.18) are fulfilled automatically [28]. This fact was proved earlier by another method in [29].…”

Section: Introductionsupporting

confidence: 57%

“…The proof is based on the K. Veselić criterion of regularity [25,26] adapted to the case of definitizable operators in [27]. In the case when A + and A − are Hilbert space symmetric operators, similar results were obtained in [23] and [28].…”

Section: Introductionmentioning

confidence: 90%

“…A motivation for this name is hidden in the fact, that the mapping f → f is a unitary mapping from K to a reproducing kernel Kreȋn space with the kernel M (z) − M (w) z −w (see [28] for the Hilbert space case). For every z ∈ ρ(A 0 ), the following equivalence holds:…”

Section: Definition 42mentioning

confidence: 99%

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Coupling of definitizable operators in Krein spaces

Derkach¹,

Trunk²

2017

Nanosystems: Phys. Chem. Math.

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Indefinite Sturm-Liouville operators defined on R are often considered as a coupling of two semibounded symmetric operators defined on R + and R − , respectively. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension.In this paper, we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm-Liouville problems on R.

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Section: Introductionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 90%

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Coupling of definitizable operators in Krein spaces

Derkach¹,

Trunk²

2017

Nanosystems: Phys. Chem. Math.

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“…The following theorem is the indefinite version of a result from [19] (presented in this form in [11,23]). Theorem 3.1.…”

Section: Regularity Of Critical Points Of Couplings In Kreȋn Spacesmentioning

confidence: 95%

“…An important ingredient of the coupling method is the boundary triples technique (see [29] and references therein) and the theory of the corresponding abstract Weyl functions developed in [20]. Indefinite Sturm-Liouville operators were studied by the coupling method in [38,41,39,40,11,23]. In particular, a criterion of regularity of critical points 0 and ∞ in the case when |w| and r are even was found in [46].…”

mentioning

confidence: 99%

Indefinite Sturm-Liouville operators in polar form

Ćurgus¹,

Derkach²,

Trunk³

2021

Preprint

Self Cite

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Let B + and B − be nonnegative symmetric operators with defect numbers (1, 1) in Hilbert spaces H + and H − . We consider the coupling A of operators A + = B + and A − = −B − which is self-adjoint, nonnegative and has a nonempty resolvent set in the Kreȋn space K with the fundamental decomposition K = H + [+]H − . Such an operator is definitizable and has at most two critical points at ∞ and 0. It is similar to a self-adjoint operator in a Hilbert space if and only if its critical points are regular and there are no Jordan chains at 0. We find criteria for the regularity of critical points of A formulated in terms of the abstract Weyl functions m + and m − of the operators B + and B − .A typical example of such coupling is the operator A generated in the Kreȋn space K = L 2 w (I) by the differential expression> 0 and (sgn t)w(t) > 0 a.e. on I. The operator A is self-adjoint in the Kreȋn space L 2 w (I) and can be considered as the coupling of the restrictions of A onto L 2 w − (I − ) and L 2 w + (I + ), where I − = (b − , 0), I + = (0, b + ), w − is the restriction of −w onto I − and w + is the restriction of w onto I + . For this operator the general criteria for the regularity of a critical point are reformulated in terms of the coefficients w and r.

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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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Partially Fundamentally Reducible Operators in Kreĭn Spaces

Cited by 4 publications

References 40 publications

Coupling of definitizable operators in Krein spaces

Coupling of definitizable operators in Krein spaces

Indefinite Sturm-Liouville operators in polar form

Realizations of Meromorphic Functions of Bounded Type

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