Self-adjoint operators with inner singularities and pontryagin spaces

Aad Dijksma, Heinz Langer, Yuri Shondin, Chris Zeinstra

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    Let A(0) be an unbounded self-adjoint operator in a Hilbert space H-0 and let chi be a generalized element of order -m -1 in the rigging associated with Ag and the inner product (., .)(0) of H-0. In [S1, S2, S3] operators H-t, t epsilon R U infinity, are defined which serve as an interpretation for the family of operators A(0) + t(-1)(. , chi)(0) chi. The second summand here contains the inner singularity mentioned in the title. The operators H-t act in Pontryagin spaces of the form Pi(m) = H(0)circle plus C-m circle plus C-m where the direct summand space C-m circle plus C-m is provided with an indefinite inner product. They can be interpreted both as a canonical extension of some symmetric operator S in Pi(m) and also as extensions of a one-dimensional restriction S-0 of A(0) in H-0 and hence they can be characterized by a class of Straus extensions of S-0 as well as via M.G. Krein's formulas for (generalized) resolvents. In this paper we describe both these realizations explicitly and study their spectral properties. A main role is played by a special class of Q-functions. Factorizations of these functions correspond to the separation of the nonpositive type spectrum from the positive spectrum of H-t. As a consequence, in Subsection 7.3 a family of self-adjoint Hilbert space operators is obtained which can serve as a nontrivial quantum model associated with the operators Ag + t(-1)(. , chi)(0) chi.
    Original languageEnglish
    Title of host publicationOperator Theory and Related Topics
    Subtitle of host publicationProceedings of the Mark Krein International Conference on Operator Theory and Applications, Odessa, Ukraine, August 18–22, 1997 Volume II
    EditorsV.M. Adamyan, I. Gohberg, M. Gorbachuk, V. Gorbachuk, M.A. Kaashoek, H. Langer, G. Popov
    Number of pages71
    ISBN (Print)3-7643-6288-X
    Publication statusPublished - 2000

    Publication series

    NameOperator Theory: Advances and Applications
    ISSN (Print)0255-0156




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