The matrix analysis of spectral projections for the perturbed self-adjoint operators

We study bounded perturbations of an unbounded positive definite self-adjoint operator with discrete spectrum. The spectrum has semi-simple eigenvalues with finite geometric multiplicity and the perturbation belongs to operator space defined by rate of the off-diagonal decay of the operator matrix. We show that the spectral projections and the resolvent of the perturbed operator belong to the same space as the perturbation. These results are applied to the Hill operator and the operator with matrix potential. We also consider the inverse problem and the modified Galerkin method.