The approached calculation of eigenvalues of the discrete operator by means of spectral traces of resolvent degrees

Let a discrete self-adjoint operator $T$ acts in a separable Hilbert space and have the kernel resolvent, and eigenvalues and eigenfunctions of the operator $T$ be known. In the paper the method of calculation of eigenvalues of the perturbed operator $T+P$ is considered. Resolvent of this operator is presented as convergent Neumann series on eigenfunctions of the operator $T$. The point of the method is that at first is found a set of numbers which approximate traces of the resolvent degrees of the operator $T+P$. Then by means of the given set, the system of nonlinear algebraic equations is constructed and solved. The solution of the systemis a set of numbers which approximate first eigenvalues of the resolvent of the perturbed operator $T+P$.