Abstract
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.
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References
T.Ja. Azizov, Dissipative operators in Hilbert space with indefinite metric, Izv. Acad. Nauk SSSR Ser. Mat. 37 (1973), no. 3 (Russian); English trans, in Math USSR Izv. 7 (1973).
T. Ja. Azizov and I.S. Iohvidov, Linear operators in spaces with indefinite metric, John Wiley, Chichester, 1989.
L.Barkwell, P.Lancaster, and A.S.Markus, Gyroscopically stabilized systems: a class of quadratic eigenvalue problems with real spectrum, Canadian J.Math. 44 (1992), 42–53.
N.G. Chetaev, The stability of motion, Pergamon Press, 1961.
A.M. Gomilko, Invariant subspaces of J-dissipative operators, J. Funct. Anal, and Appl. 19 (1985), no. 3, 213–214.
I. Gohberg, P. Lancaster and L. Rodman, Matrices and indefinite scalar product, Operator theory: Advances and Applications, Vol. 8, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1983.
I. Gohberg and E.Sigal, An Operator Generalization of the Logarithmic Residue Theorem and the Theorem of Rouché, Mat. Sbornik 84 (1971); English transl. in Math. USSR Sbornik 13 (1971), 603–625.
R.O.Griniv, On operator pencils arising in the problem of semiinfinite beam oscillations with internal damping, Moscow Univ. Math. Bulletin (to appear).
T. Kato, Perturbation theory for linear operators (2-nd edition), Springer-Verlag, New York, 1976.
M.V. Keldysh, On the completeness of eigenfunctions of certain classes of nonselfadjoint linear operators., Russian Math. Surveys 26 (1971), no. 4, 295–305.
A.G. Kostyuchenko amd M.B. Orazov, On certain properties of the roots of a selfadjoint quadratic pensil, J. Funct. Anal Appl. 9 (1975), 28–40.
A.G. Kostyuchenko and A.A. Shkalikov, Selfadjoint quadratic operator pencils and elliptic problems, J. Funct. Anal, and Appl. 17 (1983), 109–128.
M.G.Krein and H.Langer, On Definite Subspaces and Generalized Resolvents of Hermitian Operators in Spaces Π k , Funkz. Anal, i Prilozh. vol 5 (1971), no. 2, 59–71; vol 5 (1971), no. 3, 54–69 (Russian); English transl in Funct. Anal, and Appl. 5 (1971).
W. Tompson (Lord Kelvin) and P. Tait, Treatise on Natural Philosophy, Part 1, Cambrige Univ. Press, 1869.
J.L. Lions and E.Magenes, Problems aux Limites Nonhomogenes et Applications. Vol. 1, Dunod, Paris, 1968; English transl. in Springer Verlag, 1972.
P. Lancaster and A.A. Shkalikov, Damped vibrations of beams and related spectral problems, Can. Appl. Math. Quart. 2 (1994), no. 1, 45–90.
P. Lancaster and M. Tismenetsky, Inertia characteristics of selfadjoint matrix polynomials, Lin. Algebra and Appl. 52/53 (1983), 479–496.
A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Soc., Providence, 1988.
A.I. Miloslavskii, Foundation of the spectral approach in nonconservative problems of the theory of elastic stability, J. Funct. Anal. Appl. 17 (1983), no. 3, 233–235.
—, On stability of some classes of evolutionary equations, Siberian Math. J. 26 (1985), no. 5, 723–735.
L.S. Pontrjagin, Hermitian operators in spaces with indefinite metric, Izv. Acad. Nauk SSSR Ser. Mat. 8 (1944), 243–280. (Russian)
M.P. Paidoussis and N.T. Issid, Dynamic stability of pipes conveying fluid, J. Sound Vibration 33 (1974), 267–294.
V.N. Pivovarchik, A boundary value problem connected with the oscillation of elastic beams with internal and viscous damping, Moscow Univ. Math. Bulletin 42 (1987), 68–71.
—, On oscillations of a semiinfinite beam with internal and external damping, Prikladnaya Mathem. and Mech. 52 (1988), no. 5, 829–836 (Russian); English transl. in J. Appl. Math. and Mech. (1989).
—, On the spectrum of quadratic operator pencils in the right half plane, Matem. Zametki 45 (1989), no. 6, 101–103 (Russian); English transl. in Math. Notes 45 (1989).
—, On the total algebraic multiplicity of spectrum in the right half plane for one class of quadratic operator pencils, Algebra and Analysis 3 (1991), no. 2, 223–230.
A.C.M. Ran and D. Temme, Dissipative matrices and invariant maximal semidefinite subspaces, Linear Algebra Appl. (to appear).
A. A. Shkalikov, Selection principles and properties of some parts of eigen and associated elements of operator pencils, Moscow Univ. Math. Bulletin 43 (1988), no. 4, 16–25.
—, Operator pencils and operator equations in Hilbert space, (Unpublished manuscript, University of Calgary), 1992.
—, Elliptic equations in Hilbert space and associated spectral problems, J. Soviet Math. 51 (1990), no. 4, 2399–2467.
A.A. Shkalikov and R.O. Griniv, On operator pencils arising in the problem of beam oscillation with internal damping, Matem. Zametkii 56 (1994), no. 2, 114–131 (Russian); English transl. in Math. Notes 56 (1994).
H.K. Wimmer, Inertia theorems for matricies, controllability and linear vibrations, Linear Algebra Appl. (1974), no. 8, 337–343.
E.E. Zajac, The Kelvin-Tait-Chetaev theorem and extentions, J. Aeronaut. Sci. vol 11 (1964), no. 2, 46–49.
V.N. Zefirov, V.V. Kolesov and A.I. Miloslavskii, On eigenfrequences of a strightline pipe, Izv. Acad. Nauk SSSR, Ser. Mech. Tverdogo Tela (1985), no. 1, 179–188 (Russian); English transl. in Math. USSR Izv. Ser. Mech (1985).
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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
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