Abstract
Self-adjoint quadratic operator pencilsL(λ)=λ 2 A + λB + C with a noninvertible leading operatorA are considered. In particular, a characterization of the spectral points of positive and of negative type ofL is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[Ba]Barston, E.M.: A minimax principle for nonoverdamped systems, Int. J. Engng. Sci. 12 (1974), 413–421.
[BEL]Binding, P., Eschwé, J., Langer, H.: Variational principles for real eigenvalues of self-adjoint operator pencils, Integral Equations Operator Theory (to appear)
[GLaR]Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials, Academic Press, New York, 1982.
[KL 1]Krein, M.G., Langer, H.: On some mathematical principles in the linear theory of damped oscillations of continua, Proc. Int. Sympos. on Applications of the Theory of Functions in Continuum Mechanics, Tbilissi, 1963, Vol II: Fluid and Gas Mechanics, Math. Methods, Moscow, 1965, 283–322 (Russian); English transl: Integral Equations Operator Theory 1 (1978), 364–399, 539–566.
[KL 2]Krein, M.G., Langer, H.: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raum ∏κ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236.
[Kr]Krupnik, I.: Defect basis property of the eigenvectors of a selfadjoint operator-valued function and formula for defect, Integral Equations Operator Theory 16 (1993), 131–144.
[LaMMa]Lancaster, P., Markus, A.S., Matsaev, V.I.: Definitizable operators and quasihyperbolic operator polynomials, J. Funct. Anal. 131 (1995), 1–28.
[L1]Langer, H.: Über eine Klasse polynomialer Scharen selbstadjungierter Operatoren im Hilbertraum, J. Functional Analysis 12 (1973), 13–29.
[L2]Langer, H.: Spectral functions of definitizable operators in Kreîn spaces, Proc. Graduate School “Functional Analysis”, Dubrovnik 1981. Lecture Notes in Math. 948, Springer Verlag, Berlin, 1982, 1–46.
[Li]Lifschitz, A.E.: Magnetohydrodynamics and Spectral Theory, Kluwer Academic Publishers, Dordrecht, Boston, London, 1989.
[LMMa 1]Langer, H., Markus, A., Matsaev, V.: Locally definite operators in indefinite inner product spaces, Math. Ann. 308 (1997), 405–424.
[LMMa 2]Langer, H., Markus, A., Matsaev, V.: Linearization and compact perturbation of analytic operator functions, Operator Theory: Advances and Applications (to appear)
[LMT]Langer, H., Markus, A., Tretter, C.: Corners of numerical ranges, Operator Theory: Advances and Applications (to appear)
[M]Markus, A.: Introduction to the spectral theory of polynomial operator pencils, Amer. Math. Soc., Transl. Math. Monographs 71, 1988.
[MMa]Markus, A., Matsaev, V.: Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone, Integral Equations Operator Theory 16 (1993), 539–564.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Adamjan, V., Langer, H. & Möller, M. Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils. Integr equ oper theory 39, 127–152 (2001). https://doi.org/10.1007/BF01195813
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01195813