On the diagonalizability conditions of a perturbed difference operator in some spaces

The current paper uses the method of similar operators to obtain the conditions for reducing the matrix of a difference operator of the form $(Ax)(n)=an x(n)-c_1(n)x(n-1)-c_2(n)x(n+1)$ to a diagonal or block-diagonal form in the standard basis of space $\ell_2$ and estimates its spectral characteristics. The paper presents the main definitions of the method used. In accordance with the method of such operators, the operator $A$ is represented in the form $A=A_0-B$ where the matrix operator $A_0$ has a diagonal structure, and $B$ is the perturbation operator. The conditions for the perturbation operator $B$ are considered in cases when this operator belongs to three different spaces. The asymptotic representation of eigenvalues, estimates of eigenvectors, and elements of matrices of spectral projectors are also obtained.