
This article is cited in 2 scientific papers (total in 2 papers)
Short Notes
The spectral identity for the operator with nonnuclear resolvent
E. V. Kirillov^{} ^{} South Ural State University (Chelyabinsk, Russian Federation)
Abstract:
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 nonnuclear 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 nonnuclear resolvent. In this paper, we consider a direct spectral problem for an operator with a nonnuclear 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 nonnuclear 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 nonnuclear resolvent.
Keywords:
perturbed operator, discrete selfadjoint operator, direct spectral problem, relative resolvent.
DOI:
https://doi.org/10.14529/jcem170107
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UDC:
517.9
MSC: 35A01, 35E15, 35Q19 Received: 02.03.2017
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Citation:
E. V. Kirillov, “The spectral identity for the operator with nonnuclear resolvent”, J. Comp. Eng. Math., 4:1 (2017), 69–75
Citation in format AMSBIB
\Bibitem{Kir17}
\by E.~V.~Kirillov
\paper The spectral identity for the operator with nonnuclear resolvent
\jour J. Comp. Eng. Math.
\yr 2017
\vol 4
\issue 1
\pages 6975
\mathnet{http://mi.mathnet.ru/jcem85}
\crossref{https://doi.org/10.14529/jcem170107}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3637690}
\elib{http://elibrary.ru/item.asp?id=28921547}
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This publication is cited in the following articles:

E. V. Kirillov, G. A. Zakirova, “A direct spectral problem for $L$spectrum of the perturbed operator with a multiple spectrum”, J. Comp. Eng. Math., 4:3 (2017), 19–26

E. V. Kirillov, G. A. Zakirova, “Spectral problem for a mathematical model of hydrodynamics”, J. Comp. Eng. Math., 5:1 (2018), 51–56

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