Boundary relations and generalized resolvents of symmetric operators View Full Text


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Article Info

DATE

2009-03

AUTHORS

V. Derkach, S. Hassi, M. Malamud, H. de Snoo

ABSTRACT

The Kreĭn-Naĭmark formula provides a parametrization of all selfadjoint exit space extensions of a (not necessarily densely defined) symmetric operator in terms of maximal dissipative (in ℂ+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct generalized resolvents from given parameter families. A general version of the coupling method is introduced and the role of the boundary relations and their Weyl families in the Kreĭn-Naĭmark formula is investigated and explained. These notions lead to several new results and new types of solutions to problems involving generalized resolvents and their applications, e.g., in boundary-value problems for (ordinary and partial) differential operators. For instance, an old problem going back to M. A. Naĭmark and concerning the analytic characterization of the so-called Naĭmark extensions is solved. More... »

PAGES

17-60

References to SciGraph publications

  • 1992-12. On a formula of the generalized resolvents of a nondensely defined Hermitian operator in UKRAINIAN MATHEMATICAL JOURNAL
  • 1988. Solvable Models in Quantum Mechanics in NONE
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  • 1974. Monotone Matrix Functions and Analytic Continuation in NONE
  • 2007. Generalized Resolvents of a Class of Symmetric Operators in Krein Spaces in OPERATOR THEORY IN INNER PRODUCT SPACES
  • 2002-09. Weyl function and spectral properties of self-adjoint extensions in INTEGRAL EQUATIONS AND OPERATOR THEORY
  • 2007-11. Boundary Relations and Generalized Resolvents of Symmetric Operators in Krein Spaces in INTEGRAL EQUATIONS AND OPERATOR THEORY
  • 2006-08. Spectra of Schrödinger Operators on Equilateral Quantum Graphs in LETTERS IN MATHEMATICAL PHYSICS
  • 1998. The sum of matrix nevanlinna functions and self-adjoint extensions in exit spaces in RECENT PROGRESS IN OPERATOR THEORY
  • 1988. Hamiltonian Systems with Eigenvalue Depending Boundary Conditions in CONTRIBUTIONS TO OPERATOR THEORY AND ITS APPLICATIONS
  • 2006-08. Boundary Value Problems with Local Generalized Nevanlinna Functions in the Boundary Condition in INTEGRAL EQUATIONS AND OPERATOR THEORY
  • 1989-10. Extension theory for symmetric operators and boundary value problems for differential equations in UKRAINIAN MATHEMATICAL JOURNAL
  • 1995-01. The extension theory of Hermitian operators and the moment problem in JOURNAL OF MATHEMATICAL SCIENCES
  • 1992-04. Characteristic functions of almost solvable extensions of Hermitian operators in UKRAINIAN MATHEMATICAL JOURNAL
  • 1972. Symmetric relations on a Hilbert space in CONFERENCE ON THE THEORY OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1134/s1061920809010026

    DOI

    http://dx.doi.org/10.1134/s1061920809010026

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1020810486


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