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.