Boundary triplets and Weyl functions. Recent developments

Derkach, Vladimir; Hassi, Seppo; Malamud, M. M.; Snoo, H.S.V. de

doi:10.1017/cbo9781139135061.008

Cited by 25 publications

(78 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…3.9]), expressing global objects such as the spectrum or resolvent in terms of local building blocks if the space can be decoupled according to a graph structure (see also the example in Section 6.5). -Given a form-valued Nevanlinna function (−l z ) z∈C\ [0,∞) , reconstruct the boundary pair up to unitary equivalence in the spirit of [55] and [30] (uniqueness will only be possible if H ∩ H D is simple, i.e., if H ∩ H D has no non-trivial self-adjoint part). -Relate the concept with the theory of Dirichlet forms (cf.…”

Section: Purpose Of This Article and Outlookmentioning

confidence: 99%

“…We will provide the proofs in a forthcoming publication, see . For a recent treatment of (quasi‐)boundary triples and related concepts we refer to the nice surveys and and the references therein (see also , , , , , , ). Note that the following concepts are formulated on the operator level: Definition Let

H^{min}

be a closed, densely defined and symmetric operator in

scriptH

and set

H^{trueprefixmax} = {(H^{trueprefixmin})}^{*}

.…”

Section: Introductionmentioning

confidence: 99%

“…In , the more general notation of boundary relation has been introduced (basically, multi‐valued operators and boundary maps are allowed). One of the results of states a characterisation of boundary relations by their Nevanlinna “functions”

- Λ (\cdot)

(see also the survey ). In particular, generalised boundary triples have a Nevanlinna function

- Λ (\cdot)

with values in

B (G)

(the space of bounded operators) such that

- Im Λ (z)

is injective , while ordinary boundary triples have again values in

B (G)

and uniformly positive operators

- Im Λ (z)

(so‐called uniformly strict Nevanlinna functions), see for more such characterisations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Boundary pairs associated with quadratic forms

Post

2015

Math. Nachr.

View full text Add to dashboard Cite

We introduce a purely functional analytic framework for elliptic boundary value problems in a variational form. We define abstract Neumann and Dirichlet boundary conditions and a corresponding Dirichlet‐to‐Neumann operator, and develop a theory relating resolvents and spectra of these operators. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non‐smooth) boundary, the Zaremba (mixed boundary conditions) problem and discrete Laplacians.

show abstract

Section: Purpose Of This Article and Outlookmentioning

confidence: 99%

H^{min}

be a closed, densely defined and symmetric operator in

scriptH

and set

H^{trueprefixmax} = {(H^{trueprefixmin})}^{*}

.…”

Section: Introductionmentioning

confidence: 99%

- Λ (\cdot)

(see also the survey ). In particular, generalised boundary triples have a Nevanlinna function

- Λ (\cdot)

with values in

B (G)

(the space of bounded operators) such that

- Im Λ (z)

is injective , while ordinary boundary triples have again values in

B (G)

and uniformly positive operators

- Im Λ (z)

(so‐called uniformly strict Nevanlinna functions), see for more such characterisations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Boundary pairs associated with quadratic forms

Post

2015

Math. Nachr.

View full text Add to dashboard Cite

show abstract

“…On the other hand, the notion of quasi boundary triple is a particular embodiment of the more general notion of isometric/unitary boundary triples. The latter goes back to Calkin [13] and was studied in detail in [15,17].…”

Section: Quasi Boundary Triples and Their Weyl Functionsmentioning

confidence: 99%

Coupling of symmetric operators and the third Green identity

et al. 2017

View full text Add to dashboard Cite

show abstract

“…More information concerning different generalizations of the notion of boundary triplet can be found in [39].…”

Section: Definition 4 ([37]mentioning

confidence: 99%

Boundary Triplets, Weyl Functions, and the Kreĭn Formula

Derkach

2014

Operator Theory

View full text Add to dashboard Cite

This chapter contains a short review of the theory of boundary triplets, and the corresponding Weyl functions, of symmetric operators in Hilbert and Kreȋn spaces. The theory of generalized resolvents of such operators is exposed from the point of view of boundary triplets approach. Applications to different continuation problems related to the extension theory of Kreȋn space symmetric operators are discussed.

show abstract

Boundary triplets and Weyl functions. Recent developments

Cited by 25 publications

References 53 publications

Boundary pairs associated with quadratic forms

Boundary pairs associated with quadratic forms

Coupling of symmetric operators and the third Green identity

Boundary Triplets, Weyl Functions, and the Kreĭn Formula

Contact Info

Product

Resources

About