Boundary relations and generalized resolvents of symmetric operators

Derkach, Vladimir; Hassi, Seppo; Malamud, M. M.; Snoo, Henk de

doi:10.1134/s1061920809010026

Cited by 94 publications

(147 citation statements)

References 33 publications

Supporting

Mentioning

144

Contrasting

Unclassified

Order By: Relevance

“…2) In the case H 1 = H 0 =: H the class R(H) := R(H, H) coincides with the known class of Nevanlinna functions τ (·) : C \ R → C(H) (see for instance [6]) and (2.13) takes the form 16) where…”

Section: Holomorphic Operator Pairs Recall That a Holomorphic Operatmentioning

confidence: 91%

“…More precisely this means that the collection Π = {H, Γ 0 , Γ 1 } is a boundary triplet (boundary value space) for A * in the sense of [9]. In this case the relationsObserve also that for a boundary triplet Π Proposition 2.6 follows from the Π-admissibility criterion obtained in [5,6].be an interval on the real axis (in the case b < ∞ the point b may or may not belong to ∆), let H be a separable Hilbert space and letbe a differential expression of an even order 2n with smooth enough operator-valued. Denote by y [k] (·), k = 0 ÷ 2n the quasi-derivatives of a vector-function y(·) : ∆ → H, corresponding to the expression (2.26) and let D(l) be the set of functions y(·) for which this expression makes sense [20,21,22].…”

mentioning

confidence: 99%

See 1 more Smart Citation

Minimal spectral functions of an ordinary differential operator

Mogilevskii¹

2012

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

Abstract. Let l[y] be a formally selfadjoint differential expression of an even order on the interval [0, b (b ≤ ∞) and let L0 be the corresponding minimal operator. By using the concept of a decomposing boundary triplet we consider the boundary problem formed by the equation l[y] − λy = f (f ∈ L2[0, b ) and the Nevanlinna λ-depending boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the m-function, which in the case of selfadjoint decomposing boundary conditions coincides with the classical characteristic (Titchmarsh-Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e., all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indices n±(L0) and not decomposing boundary conditions) the known estimate of the spectral multiplicity of the (exit space) selfadjoint extension A ⊃ L0. The results of the paper are obtained for expressions l[y] with operator valued coefficients and arbitrary (equal or unequal) deficiency indices n±(L0). IntroductionThe main objects of the paper are differential operators generated by a formally selfadjoint differential expression l[y] of an even order 2n on an interval ∆ = [0, b (b ≤ ∞). We consider the expression l[y] with operator valued coefficients and arbitrary (possibly unequal) deficiency indices, but in order to simplify presentation of the main results assume thatis a scalar expression with real-valued coefficients p k (t) (t ∈ ∆) [20]. Denote by L 0 and L(= L * 0 ) minimal and maximal operators respectively generated by the expression (1.1) in the Hilbert space H := L 2 (∆) and let D be the domain of L. As is known L 0 is a symmetric operator with equal deficiency indices m = n ± (L 0 ) and n ≤ m ≤ 2n. Denote also by n b := m − n the defect number of the expression (1.1) at the point b [17].2000 Mathematics Subject Classification. 34B05, 34B20, 34B40, 47E05. Key words and phrases. Differential operator, decomposing D-boundary triplet, boundary conditions, minimal spectral function, spectral multiplicity. In the present paper we develop an approach based on the concept of a decomposing boundary triplet for a differential operator [17,18,19]. Recall that according to [17] a decomposing boundary triplet for L is a boundary triplet Π = {C n ⊕ C n b , Γ 0 , Γ 1 } in the sense of [9] with the boundary operators Γ j : D → C n ⊕ C n b , j ∈ {0, 1} of the special formHere y (j) (0) are vectors of quasi-derivatives (2.27) at the point 0 and Γ ′ j y(∈ C n b ), j ∈ {0, 1} are vectors of boundary values of a function y ∈ D at the singular endpoint b.Next assume that P = {C 0 (λ), C 1 (λ)} (λ ∈ C \ R) is a Nevanlinna operator pair defined by the block representations3) with the constant entriesĈ 0 ,Ĉ 1 and let τ = τ (λ) := {{h, h ′ } : C 0 (λ)h+C 1 (λ)h ′ = 0} be the corresponding Nevanlinna family of linear relations. Denote byK the range of the operatorĈ = (Ĉ 0Ĉ1 ) and let n = dimK = rank(Ĉ 0Ĉ1 ), n ′ = m −n.Th...

show abstract

Section: Holomorphic Operator Pairs Recall That a Holomorphic Operatmentioning

confidence: 91%

mentioning

confidence: 99%

Minimal spectral functions of an ordinary differential operator

Mogilevskii¹

2012

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Our standard references for the theory of boundary triples are [DHMdS06] and the survey article [DHMdS09]. There also some basic notations and results about linear relations can be found.…”

Section: Boundary Relationsmentioning

confidence: 99%

Spectral Multiplicity of Selfadjoint Schrödinger Operators on Star-Graphs with Standard Interface Conditions

Simonov¹,

Woracek

2013

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schrödinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of a Theorem of I.S.Kac.

show abstract

“…For an account on boundary control systems dealt with in the literature, we refer the reader to [1,15,16,22,29,31,34,35], where also strategies from the theory of selfadjoint extensions of symmetric operators come into play, [3,4,7,28,26]. As a first illustrative example of boundary control systems we discuss in Subsection 5.1 the notion of port-Hamiltonian systems as introduced in [10], also see [9].…”

Section: Introductionmentioning

confidence: 99%

“…In the literature, in order to discuss boundary control systems in an operatortheoretic framework, the concept of boundary triples is used, see e.g. [16,3,4,7], we also refer to [26,28], where in [26] a unified perspective is given. A boundary triple is a symmetric operator S defined in a Hilbert space H and two continuous linear operators…”

mentioning

confidence: 99%

On a comprehensive class of linear control problems

Picard

Trostorff

Waurick

2014

IMA J Math Control Info

View full text Add to dashboard Cite

Abstract. We discuss a class of linear control problems in a Hilbert space setting. This class encompasses such diverse systems as port-Hamiltonian systems, Maxwell's equations with boundary control or the acoustic equations with boundary control and boundary observation. The boundary control and observation acts on abstract boundary data spaces such that the only geometric constraint on the underlying domain stems from requiring a closed range constraint for the spatial operator part, a requirement which for the wave equation amounts to the validity of a Poincare-Wirtinger-type inequality. We also address the issue of conservativity of the control problems under consideration.

show abstract

Boundary relations and generalized resolvents of symmetric operators

Cited by 94 publications

References 33 publications

Minimal spectral functions of an ordinary differential operator

Minimal spectral functions of an ordinary differential operator

Spectral Multiplicity of Selfadjoint Schrödinger Operators on Star-Graphs with Standard Interface Conditions

On a comprehensive class of linear control problems

Contact Info

Product

Resources

About