Standard symmetric operators in Pontryagin spaces: a generalized von Neumann formula and minimality of boundary coefficients

Azizov, T. Ya.; Ćurgus, Branko; Dijksma, Aad

doi:10.1016/s0022-1236(02)00041-1

Cited by 12 publications

(11 citation statements)

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“…Here and below, by a subspace we mean a closed linear subset. We apply the result to the following cases: 𝑇 is a standard relation [3] Conditions for the equality in 𝑀 Γ (𝑧) ⊆ 𝑀 Γ (𝑧) to hold are also discussed. The topic is important, for example, in a Hilbert space setting, provided one asks whether or not the Weyl family 𝑀 Γ corresponding to an e.u.b.p.…”

Section: Synopsis Main Resultsmentioning

confidence: 99%

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Weyl families of transformed boundary pairs

Jursenas

2023

Mathematische Nachrichten

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Let be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space . Let be the Weyl family corresponding to . We cope with two main topics. First, since need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation , for some , becomes a nontrivial task. Regarding as the (Shmul'yan) transform of induced by Γ, we give conditions for the equality in to hold and we compute the adjoint . As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for is nonempty. Based on the criterion for the closeness of , we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family corresponding to a boundary relation Γ for is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair with its Weyl family . The transformation scheme is either or with suitable linear relations V. Results in this direction include but are not limited to: a 1‐1 correspondence between and ; the formula for , for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple with and (second scheme, Hilbert space case).

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Section: Synopsis Main Resultsmentioning

confidence: 99%

“…Here and below, by a subspace we mean a closed linear subset. We apply the result to the following cases: T is a standard relation [3] in a Pontryagin space (Corollary 4.6, Examples 4.2, 5.1) and

\Gamma =\overline{\Gamma }

is a b.r . for

T^+

, in which case

M_\Gamma

is a generalized Nevanlinna family (cf.…”

Section: Introductionmentioning

confidence: 99%

Weyl families of transformed boundary pairs

Jursenas

2023

Mathematische Nachrichten

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“…If additionally, α = β then also (1.5) The results in this note generalize and complete some algebraic ones in [2]. The notion of standard symmetric linear relation has been introduced in [2], and in connection with this notion, a von Neumann like formula has been proved for the case of standard symmetric linear operators.…”

Section: Introductionmentioning

confidence: 58%

“…The notion of standard symmetric linear relation has been introduced in [2], and in connection with this notion, a von Neumann like formula has been proved for the case of standard symmetric linear operators. The main ingredients of the proof of that formula are some descriptions of the range and the kernel of certain polynomial with simple squares in a linear operator.…”

Section: Introductionmentioning

confidence: 99%

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Some Basic Properties of Polynomials in a Linear Relation in Linear Spaces

Sandovici

Operator Theory in Inner Product Spaces

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Boundary value problems with eigenvalue depending boundary conditions

Behrndt

2009

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Key words Boundary value problem, locally definitizable operator and relation, locally definitizable function, boundary triplet, Weyl function, Krein space MSC (2000) 47B50, 34B07, 46C20, 47A06, 47B40We investigate some classes of eigenvalue dependent boundary value problems of the formwhere A ⊂ A + is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ0, Γ1 are abstract boundary mappings. It is assumed that A admits a self-adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self-adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self-adjoint extension A of A × T in K × H with the property that the compressed resolvent PK A − λ ¡ −1 ¬ ¬ K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so-called linearization A are studied. The general results are applied to indefinite Sturm-Liouville operators with eigenvalue dependent boundary conditions.

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Standard symmetric operators in Pontryagin spaces: a generalized von Neumann formula and minimality of boundary coefficients

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Cited by 12 publications

References 8 publications

Weyl families of transformed boundary pairs

Weyl families of transformed boundary pairs

Some Basic Properties of Polynomials in a Linear Relation in Linear Spaces

Boundary value problems with eigenvalue depending boundary conditions

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