On the negative squares of a class of self-adjoint extensions in Krein spaces

Authors

  • Jussi Behrndt,

    Corresponding author
    1. Technische Universität Graz, Institut für Numerische Mathematik, Steyrergasse 30, A-8010 Graz, Austria. Phone: +43 (0)316 8738127, Fax: +43 (0)316 8738621
    • Technische Universität Graz, Institut für Numerische Mathematik, Steyrergasse 30, A-8010 Graz, Austria. Phone: +43 (0)316 8738127, Fax: +43 (0)316 8738621
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  • Annemarie Luger,

    1. Department of Mathematics, Stockholm University, SE - 106 91 Stockholm, Sweden. Phone: +46 (0)8 164593, Fax: +46 (0)8 6126717
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  • Carsten Trunk

    1. Technische Universität Ilmenau, Institut für Mathematik, Postfach 10 05 65, 98684 Ilmenau, Germany. Phone: +49 (0)3677 69 3253, Fax: +49 (0)3677 69 3270
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Abstract

A description of all exit space extensions with finitely many negative squares of a symmetric operator of defect one is given via Krein's formula. As one of the main results an exact characterization of the number of negative squares in terms of a fixed canonical extension and the behaviour of a function τ (that determines the exit space extension in Krein's formula) at zero and at infinity is obtained. To this end the class of matrix valued equation image-functions is introduced and, in particular, the properties of the inverse of a certain equation image-function which is closely connected with the spectral properties of the exit space extensions with finitely many negative squares is investigated in detail. Among the main tools here are the analytic characterization of the degree of non-positivity of generalized poles of matrix valued generalized Nevanlinna functions and some extensions of recent factorization results.

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