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The Virozub–Matsaev Condition and Spectrum of Definite Type for Self-adjoint Operator Functions

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Abstract.

We establish sufficient conditions for the so-called Virozub–Matsaev condition for twice continuously differentiable self-adjoint operator functions. This condition is closely related to the existence of a local spectral function and to the notion of positive type spectrum. Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given.

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Authors and Affiliations

  1. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8–10, A-1040, Wien, Austria

    Heinz Langer

  2. Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, United Kingdom

    Matthias Langer

  3. Department of Mathematics, Ben-Gurion University of the Negev, IL-84105, Beer Sheva, Israel

    Alexander Markus

  4. Mathematisches Institut, Universität Bern, Sidlerstr. 5, CH-3012, Bern, Switzerland

    Christiane Tretter

Authors
  1. Heinz Langer
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  2. Matthias Langer
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  3. Alexander Markus
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  4. Christiane Tretter
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Corresponding author

Correspondence to Heinz Langer.

Additional information

Communicated by Daniel Alpay.

Dedicated to our friend and colleague Vladimir Matsaev on the occasion of his 70th birthday

Received: September 22, 2007. Accepted: September 29, 2007.

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Langer, H., Langer, M., Markus, A. et al. The Virozub–Matsaev Condition and Spectrum of Definite Type for Self-adjoint Operator Functions. Complex anal.oper. theory 2, 99–134 (2008). https://doi.org/10.1007/s11785-007-0032-z

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  • DOI: https://doi.org/10.1007/s11785-007-0032-z

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