Asymptotics of eigenvalues of infinite block matrices

The paper is devoted to determining the asymptotic behavior of eigenvalues, which is one of topical directions in studying operators generated by tridiagonal infinite block matrices in Hilbert spaces of infinite sequences with complex coordinates or, in other words, to discrete Sturm-Liouville operators. In the work we consider a class of non-self-adjoint operators with discrete spectrum being a sum of a self-adjoint operator serving as an unperturbed operator and a perturbation, which is an operator relatively compact with respect to the unperturbed operator. In order to study the asymptotic behavior of eigenvalues, in the paper we develop an adapted scheme of abstract method of similar operators. The main idea of this approach is that by means of the similarity operator, the studying of spectral properties of the original operator is reduced to studying the spectral properties of an operator of a simpler structure. Employing this scheme, we write out the formulae for the asymptotics of arithmetical means of the eigenvalues of the considered class of the operators. We note that such approach differs essentially from those employed before. The obtained general result is applied for determining eigenvalues of particular operators. Namely, we provide asymptotics for the eigenvalues of symmetric and non-symmetric tridiagonal infinite matrices in the scalar case, the asymptotics for arithmetical means of the eigenvalues of block matrices with power behavior of eigenvalues of unperturbed operator and generalized Jacobi matrices with various number of non-zero off-diagonals.