Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T06:39:23.052Z Has data issue: false hasContentIssue false
  • English
  • Français

Gyroscopically Stabilized Systems: A Class Of Quadratic Eigenvalue Problems With Real Spectrum

Published online by Cambridge University Press:  20 November 2018

Lawrence Barkwell ,
Peter Lancaster  and
Alexander S. Markus
Lawrence Barkwell
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4
Peter Lancaster
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4
Alexander S. Markus
Affiliation:
Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer Sheva, Israel
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Eigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.

Keywords

47A5615A22
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 1 , 01 December 1991 , pp. 42 - 53
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Barkwell, L. and Lancaster, P., Overdamped and gyroscopic vibrating systems, Jour, of Applied Mechanics, to appear.Google Scholar
2. Berberian, S.K., Lectures in functional analysis and operator theory. Springer-Verlag, 1974.Google Scholar
3. Duffin, R.J., A minimax theory for overdamped networks, Jour. Rat. Mech. and Anal. 4(1955), 221–233.Google Scholar
4. Gohberg, I., Lancaster, P. and Rodman, L., Matrix polynomials. Academic Press, New York, 1982.Google Scholar
5. Gohberg, I., Matrices and indefinite scalar products. Birkhàuser, Basel, 1983.Google Scholar
6. Gohberg, I. and Sigal, E.I., An operator generalization of the logarithmic residue theorem and Rouchés theorem, Math. USSR Sb. 13(1971), 603–625.Google Scholar
7. Kostyuchenko, A.G. and Shkalikov, A.A., Selfadjoint quadratic operator pencils and elliptic problems, Funct. Anal. Appl. 17(1983), 109–128.Google Scholar
8. Krein, M.G. and H., Langer, K., On the theory of quadratic pencils of selfadjoint operators, Soviet Math. Doklady 5(1964).Google Scholar
9. Krein, M.G. and H., Langer, K., On some mathematical principles in the linear theory of damped oscillations of continua Parts I & II. Int. Eq. and Oper. Theory 1(1978), 364–369.539–566.Google Scholar
10. Langer, H., Uber eine Klasse polynomialer Scharen selbstadjungierter Operatoren in Hilbertraum, I. Jour. Functional Anal. 12(1973), 13–29.Google Scholar
11. Markus, A.S., On holomorphic operator-valued functions, Dokl. Akad. Nauk SSSR 119(1958), 1099–1102.(Russian).Google Scholar
12. Markus, A.S., Introduction to the spectral theory of polynomial operator pencils. Amer. Math. Soc., Providence, 1988.Google Scholar
13. Markus, A.S. and Matsaev, V.I., On the basis property for a certain part of the eigenvectors and associated vectors of a selfadjoint operator pencil, Math. USSR Sb. 61(1988), 289–307.Google Scholar