Geodesic
The spectral identity for the operator with non-nuclear resolvent
Journal of computational and engineering mathematics, Tome 4 (2017) no. 1, pp. 69-75.

Voir la notice de l'article provenant de la source Math-Net.Ru

Direct spectral problems play an important role in many branches of science and technology. In a hight number of mathematical and physical problems is required to find the spectrum of various operators. The inverse spectral problems also have a wide range of applications. To solve them, we often find a solution to the direct problem. The method of regularized traces effectively al lows us to find the eigenvalues of the perturbed operator. This method is not feasible to the operator with a non-nuclear resolution. This is related to the selection of a special function that transforms the eigenvalues of the operator. Currently, there is an active search for methods that makes it possible to calculate the eigenvalues of a perturbed operator with a non-nuclear resolvent. In this paper, we consider a direct spectral problem for an operator with a non-nuclear resolvent perturbed by a bounded one.The method of regularized traces is chosen as the main method for solving this problem. Broadly speaking, this method can not be applied to this problem. It is impossible to take advantage of Lidsky's theorem because the operator has a non-nuclear resolvent. We proposed to introduce the relative resolvent of the operator. In this case, the operator $L$ was chosen so that the relative resolvent of the operator is a nuclear operator. As a result of applying the resolvent method to the relative spectrum of the perturbed operator, we obtain the relative eigenvalues of the perturbed operator with the non-nuclear resolvent.
Keywords: perturbed operator, discrete self-adjoint operator, direct spectral problem, relative resolvent. 
@article{JCEM_2017_4_1_a6,
     author = {E. V. Kirillov},
     title = {The spectral identity for the operator with non-nuclear resolvent},
     journal = {Journal of computational and engineering mathematics},
     pages = {69--75},
     publisher = {mathdoc},
     volume = {4},
     number = {1},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JCEM_2017_4_1_a6/}
}
TY  - JOUR
AU  - E. V. Kirillov
TI  - The spectral identity for the operator with non-nuclear resolvent
JO  - Journal of computational and engineering mathematics
PY  - 2017
SP  - 69
EP  - 75
VL  - 4
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JCEM_2017_4_1_a6/
LA  - en
ID  - JCEM_2017_4_1_a6
ER  - 
%0 Journal Article
%A E. V. Kirillov
%T The spectral identity for the operator with non-nuclear resolvent
%J Journal of computational and engineering mathematics
%D 2017
%P 69-75
%V 4
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JCEM_2017_4_1_a6/
%G en
%F JCEM_2017_4_1_a6
E. V. Kirillov. The spectral identity for the operator with non-nuclear resolvent. Journal of computational and engineering mathematics, Tome 4 (2017) no. 1, pp. 69-75. http://geodesic.mathdoc.fr/item/JCEM_2017_4_1_a6/

[1] V. A. Sadovnichii, V. V. Dubrovskii, “Ob odnoi abstraktnoi teoreme teorii vozmuschenii, o formulakh regulyarizovannykh sledov i o dzeta-funktsii operatorov”, Differents. uravneniya, 13:7 (1977), 1264–1271 | MR | Zbl

[2] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, v. 42, Inverse and Ill-Posed Problems Series, de Gruyer, 2012, 216+viii pp. | DOI | MR

[3] G. A. Zakirova, E. V. Kirillov, “L-regularized Trace of One of the Perturbed Operator”, Bulletin of Odessa National University. Mathematics and Mechanics, 18:2 (18) (2013), 7–13

[4] G. A. Sviridyuk, N. A. Manakova, G. A. Zakirova, “The Asymptotics of Eigenvalues of a Differential Operator in the Stochastic Models with "White Noise" ”, Applied Mathematical Sciences, 8:173-176 (2014), 8747–8754 | DOI

[5] G. A. Sviridyuk, N. A. Manakova, “The Dynamical Models of Sobolev Type with Showalter–Sidorov Condition and Additive “Noise””, Vestnik YuUrGU. Ser. Mat. Model. Progr., 7:1 (2014), 90–103 | DOI | Zbl

[6] I. M. Gelfand, B. M. Levitan, “Ob odnom prostom tozhdestve dlya sobstvennykh znachenii differentsialnogo operatora vtorogo poryadka”, Doklady Akademii nauk SSSR, 88:4 (1953), 593–596 | Zbl

[7] V. A. Sadovnichii, S. V. Konyagin, V. E. Podolskii, “Regulyarizovannyi sled operatora s yadernoi rezolventoi, vozmuschennogo ogranichennym”, Doklady Akademii nauk, 373:1 (2000), 26–28 | MR

[8] V. A. Sadovnichii, V. E. Podolskii, “Traces of operators with relatively compact perturbations”, Sb. Math., 193:2 (2002), 279–302 | DOI | DOI | MR | Zbl

[9] G. A. Zakirova, A. I. Sedov, “An Inverse Spectral Problem for Laplace Operator and it's Approximate Solution”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2:27 (127) (2008), 19–27 | Zbl