Approximate calculation of the first eigenvalues of a discrete operator in the case when spectral traces of powers of its resolvent are found approximately

We consider a discrete differential operator in a separable Hilbert space. Eigenvalues of this operator can be calculated with the use of spectral traces of powers of its resolvent. If the resolvent is a kernel operator, then finite-dimensional matrices are considered instead of powers of the resolvent and approximate values of spectral traces are calculated.