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Finite Rank Perturbations in Pontryagin Spaces and a Sturm–Liouville Problem with λ-rational Boundary Conditions

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Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations

Part of the book series: Operator Theory: Advances and Applications ((LOLS,volume 263))

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Abstract

For selfadjoint operators A 1 and A 2 in a Pontryagin space Π κ such that the resolvent difference of A 1 and A 2 is n-dimensional it is shown that the dimensions of the spectral subspaces corresponding to open intervals in gaps of the essential spectrum differ at most by n+2κ. This is a natural extension of a classical result on finite rank perturbations of selfadjoint operators in Hilbert spaces to the indefinite setting.With the help of an explicit operator model for scalar rational functions it is shown that the estimate is sharp. Furthermore, the general perturbation result and the operator model are illustrated with an application to a singular Sturm–Liouville problem, where the boundary condition depends rationally on the eigenparameter.

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Authors and Affiliations

  1. Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Jussi Behrndt

  2. Departamento de Matemática Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Ciudad Universitaria, Pabellón I, 1428, Buenos Aires, Argentina

    Friedrich Philipp

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  1. Jussi Behrndt
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Correspondence to Jussi Behrndt .

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Editors and Affiliations

  1. Mathematics, Chapman University Mathematics, Orange, California, USA

    Daniel Alpay

  2. Mathematisches Institut, Universität Leipzig Mathematisches Institut, Leipzig, Sachsen, Germany

    Bernd Kirstein

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Behrndt, J., Philipp, F. (2018). Finite Rank Perturbations in Pontryagin Spaces and a Sturm–Liouville Problem with λ-rational Boundary Conditions. In: Alpay, D., Kirstein, B. (eds) Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations. Operator Theory: Advances and Applications(), vol 263. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-68849-7_6

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