On the error estimates for the Rayleigh-Schrödinger series and the Kato-Rellich perturbation series

Let λ be a simple eigenvalue of a bounded linear operator T on a Banach space X, and let (Tn) be a resolvent operator approximation of T. For large n, let Sn denote the reduced resolvent associated with Tn and λn, the simple eigenvalue of Tn near λ. It is shown that under the assumption that all the spectral points of T which are nearest to λ belong to the discrete spectrum of T. This is used to find error estimates for the Rayleigh-Schrödinger series for λ and ϕ with initial terms λn and ϕn, where P (respectively, ϕn) is an eigenvector of T (respectively, Tn) corresponding to λ (respectively, λn), and for the Kato-Rellich perturbation series for PPn, where P (respectively, Pn) is the spectral projection for T (respectively, Tn) associated with λ (respectively, λn).