### A DIRECT SPECTRAL PROBLEM FOR L-SPECTRUM OF THE PERTURBED OPERATOR WITH A MULTIPLE SPECTRUM

#### Abstract

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.

#### Full Text:

PDF### Refbacks

- There are currently no refbacks.