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Operator Pencils Arising in Elasticity and Hydrodynamics: The Instability Index Formula

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Book cover Recent Developments in Operator Theory and Its Applications

Part of the book series: Operator Theory Advances and Applications ((OT,volume 87))

Abstract

The main object of the paper is a quadratic operator pencil of the form

$$ A(\!\lambda\!)=F\lambda^2 + (\!D +iG\!)\lambda + T, $$

with unbounded operator coefficients acting in Hilbert space. It is assumed that F,T are selfadjoint and boundedly invertible and D ≥ 0,G are symmetric and T-bounded. Pencils of this form arise as abstract models for concrete problems in elastisity and hydrodynamics. We investigate the relations between the classical and generalized spectra and under additional hypotheses prove the formula for the number of eigenvalues of A(λ) in the right-half plane. The proof of this formula is based on the preliminary investigation of maximal semidefinite invariant subspaces in the root subspaces corresponding to the pure imaginary eigenvalues of a dissipative operator in Krein or Pontrjagin spaces.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow, 199899, Russia

    A. A. Shkaliko

Authors
  1. A. A. Shkaliko
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Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    I. Gohberg

  2. Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1NA, Canada

    P. Lancaster

  3. Department of Applied Mathematics and Institute of Industrial Mathematical Sciences, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

    P. N. Shivakumar

Additional information

Dedicated to Professor Peter Lancaster on the occasion of his 65th birthday

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© 1996 Birkhäuser Verlag Basel/Switzerland

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Shkaliko, A.A. (1996). Operator Pencils Arising in Elasticity and Hydrodynamics: The Instability Index Formula. In: Gohberg, I., Lancaster, P., Shivakumar, P.N. (eds) Recent Developments in Operator Theory and Its Applications. Operator Theory Advances and Applications, vol 87. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9035-9_17

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  • DOI: https://doi.org/10.1007/978-3-0348-9035-9_17

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9878-2

  • Online ISBN: 978-3-0348-9035-9

  • eBook Packages: Springer Book Archive

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