Geodesic
A direct spectral problem for $L$-spectrum of the perturbed operator with a multiple spectrum
Journal of computational and engineering mathematics, Tome 4 (2017) no. 3, pp. 19-26.

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We consider a direct spectral problem for an operator having a non-nuclear resolvent and perturbed by the bounded operator with multiple spectrum. A similar problem was considered earlier for an operator with a single spectrum. The method of regularized traces is used as a method of solution. This method can not be applied directly to the problem. We propose to introduce the relative resolvent of the operator. A spectral problem of the form $(M+P)u=Lu$ is obtained. In this case, the operator $L$ is such 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 relative eigenvalues of the perturbed operator with non-nuclear resolvent.
Keywords: perturbed operator, discrete self-adjoint operator, direct spectral problem, relative resolvent, multiple spectrum. 
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E. V. Kirillov; G. A. Zakirova. A direct spectral problem for $L$-spectrum of the perturbed operator with a multiple spectrum. Journal of computational and engineering mathematics, Tome 4 (2017) no. 3, pp. 19-26. http://geodesic.mathdoc.fr/item/JCEM_2017_4_3_a2/

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