Operator Pencils Arising in Elasticity and Hydrodynamics: The Instability Index Formula
A. A. Shkaliko · Birkhäuser Basel eBooks · 1996
The main object of the paper is a quadratic operator pencil of the form 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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.