Geodesic
A theorem on comparison of spectra, and spectral asymptotics for a Keldysh pencil
Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 389-404 
A. S. Markus; V. I. Matsaev. A theorem on comparison of spectra, and spectral asymptotics for a Keldysh pencil. Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 389-404. http://geodesic.mathdoc.fr/item/SM_1985_51_2_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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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[2] Gokhberg I. D., Krsin M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[3] Matsaev V. I., Palant Yu. A., “O raspredelenii spektra polinomialnogo operatornogo puchka”, DAN Arm. SSR, 17:5 (1966), 257–261

[4] Matsaev V. I., Palant Yu. A., “O raspredelenii spektra drobno-ratsionalnogo puchka”, Matem. issledovaniya, 4:2 (1969), 156–157 | MR | Zbl

[5] Matsaev V. I., Palant Yu. A., Spektralnaya teoriya operatornykh puchkov, Izd-vo DGU, Donetsk, 1975

[6] Radzievskii G. V., “Asimptotika raspredeleniya kharakteristicheskikh chisel operator-funktsii, analiticheskikh v ugle”, Matem. sb., 112(154) (1980), 396–420 | MR | Zbl

[7] Markus A. S, Matsaev V. I., “Ob asimptotike spektra operatorov, blizkikh k normalnym”, Funkts. analiz, 13:3 (1979), 93–94 | MR | Zbl

[8] Markus A. S, Matsaev V. I., “Teoremy sravneniya spektrov lineinykh operatorov i spektralnye asimptotiki”, Trudy MMO, 1982, 45 | MR | Zbl

[9] Keldysh M. V., “O polnote sobstvennykh funktsii nekotorykh klassov nesamosopryazhennykh lineinykh operatorov”, UMN, 26:4 (1971), 15–41 | MR | Zbl

[10] Levin B. Ya., Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956

[11] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[12] Gokhberg I. D., Sigal E. I., “Operatornoe obobschenie teoremy o logarifmicheskom vychete i teoremy Rushe”, Matem. sb., 84(126) (1971), 607–629 | Zbl

[13] Gokhberg I. D., Krein M. G., “Osnovnye polozheniya o defektnykh chislakh, kornevykh chislakh i indeksakh lineinykh operatorov”, UMN, 12:2 (1957), 43–118 | MR

[14] Mogulskii E. Z., “Teorema polnoty sistemy sobstvennykh i prisoedinennykh vektorov ratsionalnogo operatornogo puchka”, Izv. AN Arm. SSR. Seriya matem., 3:6 (1968), 427–442 | MR | Zbl