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Mat. Sb. (N.S.), 1984, Volume 123(165), Number 3, Pages 391–406 (Mi msb2027)  

This article is cited in 11 scientific papers (total in 11 papers)

A theorem on comparison of spectra, and spectral asymptotics for a Keldysh pencil

A. S. Markus, V. I. Matsaev


Abstract: Suppose that $H$ is a normal operator, the pencil $L_0(\lambda)=I-\lambda^nH^n$ has a discrete and positive spectrum in the domain $\Omega(2\theta,R)=\{\lambda:\lvert\arg\lambda\rvert<2\theta, |\lambda|>R\}$, and $S(\lambda)$ is an operator-valued function that is holomorphic in $\Omega(2\theta,R)$ and small in comparison to $L_0(\lambda)$ (in a certain sense). A theorem is proved on comparison of the spectra of $L(\lambda)=L_0(\lambda)-S(\lambda)$ and $L_0(\lambda)$, i.e., on an estimate of the difference $N(r)-N_0(r)$, where $N(r)$ ($N_0(r)$) is the distribution function of the spectrum of $L(\lambda)$ ($L_0(\lambda)$) in $\Omega(\theta,\rho)$ ($\rho\geqslant R$). This result implies generalizations of theorems of Keldysh on the asymptotic behavior of the spectrum of a polynomial operator pencil.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1985, 51:2, 389–404

Bibliographic databases:

UDC: 517.984
MSC: Primary 47A10, 47A55; Secondary 47A53, 47B05, 47B10, 47B15
Received: 23.07.1981

Citation: A. S. Markus, V. I. Matsaev, “A theorem on comparison of spectra, and spectral asymptotics for a Keldysh pencil”, Mat. Sb. (N.S.), 123(165):3 (1984), 391–406; Math. USSR-Sb., 51:2 (1985), 389–404

Citation in format AMSBIB
\Bibitem{MarMat84}
\by A.~S.~Markus, V.~I.~Matsaev
\paper A~theorem on comparison of spectra, and spectral asymptotics for a~Keldysh pencil
\jour Mat. Sb. (N.S.)
\yr 1984
\vol 123(165)
\issue 3
\pages 391--406
\mathnet{http://mi.mathnet.ru/msb2027}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=735713}
\zmath{https://zbmath.org/?q=an:0603.47018|0562.47014}
\transl
\jour Math. USSR-Sb.
\yr 1985
\vol 51
\issue 2
\pages 389--404
\crossref{https://doi.org/10.1070/SM1985v051n02ABEH002865}


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    1. Rokhlin D., “On the Spectral Problem of the Theory of Tides in a Bounded Domain”, Dokl. Akad. Nauk, 353:5 (1997), 619–621  mathnet  mathscinet  isi
    2. Fiedler, B, “Large patterns of elliptic systems in infinite cylinders”, Journal de Mathematiques Pures et Appliquees, 77:9 (1998), 879  crossref  isi  elib
    3. V. M. Kaplitskii, “Asymptotic behaviour of the discrete spectrum of a quasi-periodic boundary value problem for a two-dimensional hyperbolic equation”, Sb. Math., 200:2 (2009), 215–228  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. V. I. Voititskiy, N. D. Kopachevskiy, P. A. Starkov, “Multicomponent conjugation problems and auxiliary abstract boundary-value problems”, Journal of Mathematical Sciences, 170:2 (2010), 131–172  mathnet  crossref  mathscinet
    5. V. M. Kaplitskiǐ, “Asymptotics of the distribution of eigenvalues of a selfadjoint second order hyperbolic differential operator on the two-dimensional torus”, Siberian Math. J., 51:5 (2010), 830–846  mathnet  crossref  mathscinet  isi  elib
    6. V. M. Kaplitskii, “On regulariziers of unbounded linear operators in Banach spaces”, J. Math. Sci. (N. Y.), 194:6 (2013), 651–655  mathnet  crossref  mathscinet
    7. D. A. Zakora, “Operator approach to the ilyushin model for a viscoelastic body of parabolic type”, Journal of Mathematical Sciences, 225:2 (2017), 345–381  mathnet  crossref
    8. D. A. Zakora, “Model szhimaemoi zhidkosti Oldroita”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 61, RUDN, M., 2016, 41–66  mathnet
    9. A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum”, Russian Math. Surveys, 71:5 (2016), 907–964  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. D. Zakora, “On the spectrum of rotating viscous relaxing fluid”, Zhurn. matem. fiz., anal., geom., 12:4 (2016), 338–358  mathnet  crossref  mathscinet
    11. N. D. Kopachevskii, A. R. Yakubova, “O nekotorykh zadachakh, porozhdennykh polutoralineinoi formoi”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 63, no. 2, Rossiiskii universitet druzhby narodov, M., 2017, 278–315  mathnet  crossref  mathscinet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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