{
  "id": "0904.0276",
  "version": "v2",
  "published": "2009-04-01T23:25:11.000Z",
  "updated": "2022-04-25T04:28:26.000Z",
  "title": "Linear Operators and Operator Functions Associated with Spectral Boundary Value Problems",
  "authors": [
    "Vladimir Ryzhov"
  ],
  "comment": "51 pages",
  "journal": "In: Kurasov, P. et al (eds) Analysis as a Tool in Mathematical Physics. Operator Theory: Advances and Applications, vol 276, 2020",
  "doi": "10.1007/978-3-030-31531-3_31",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "The paper develops a theory of spectral boundary value problems from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary conditions is suggested and results on its solvability complemented by representations of weak and strong solutions are obtained. Existence of a closed linear operator defined by a given boundary condition and description of its domain are studied in detail. These questions are addressed on the basis of Krein's resolvent formula derived from the explicit representations of solutions also obtained here. Usual resolvent identities for two operators associated with two different boundary conditions are written in terms of the so called M-function. Abstract considerations are complemented by illustrative examples taken from the theory of partial differential operators. Other applications to boundary value problems of analysis and mathematical physics are outlined. (Initial version title \"Spectral Boundary Value Problems and their Linear Operators\", 2009)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-01T23:25:11.000Z",
      "title": "Spectral Boundary Value Problems and their Linear Operators",
      "abstract": "The paper offers a self-consistent account of the spectral boundary value problems developed from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with a generalized boundary condition is introduced and results on its solvability complemented by representations of weak and strong solutions are obtained. The question of existence of a closed linear operator defined by a given boundary condition and description of its domain is studied in detail. This question is addressed on the basis of a version of Krein's resolvent formula derived from the obtained representations for solutions. Usual resolvent identities for two operators associated with two different boundary conditions are written in terms of the so called M-operator and closed linear operators defining these conditions. Two examples illustrate the abstract core of the paper. Other applications to the theory of partial differential operators and to the mathematical physics are outlined.",
      "comment": "38 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2022-04-25T04:28:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47F05",
      "35A05",
      "35P05",
      "35C05"
    ],
    "keywords": [
      "boundary condition",
      "closed linear operator",
      "spectral boundary value problems",
      "kreins resolvent formula",
      "usual resolvent identities"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.0276R"
    }
  }
}