The spectral identity for the operator with non-nuclear resolvent

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.