J Austral Math Soc Ser A 46 pp456--468, 1989.

On the Error Estimates for the Rayleigh-Schrödinger Series and the Kato-Rellich Perturbation Series

Rekha P. Kulkarni and Balmohan V. Limaye

Abstract

Let l be a simple eigenvalue of a bounded linear operator T on a Banach space X, and let (T_n) be a resolvent operator approximation of T. For large n, let S_n denote the reduced resolvent associated with T_n and l_n, the simple eigenvalue of T_n near l. It is shown that

sup k = 1,2,...  ||(T - Tn)Snk(T - Tn)Sn|| ||Sn||k-1 ® 0,   as n ® ¥,

under the assumption that all the spectral points of T which are nearest to l belong to the discrete spectrum of T. This is used to find error estimates for the Rayleigh-Schrödinger series for l and j with initial terms l_n and j_n, where j (respectively, j_n) is an eigenvector of T (respectively, T_n) corresponding to l (respectively, l_n), and also for the Kato-Rellich perturbation series for PP_n, where P (respectively, P_n) is the spectral projection for T (respectively, T_n) associated with l (respectively, l_n).

1980 AMS Subject Classification (1985 Revision): 41A25, 41A35, 41A65, 47A70

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Authors

Rekha P. Kulkarni

Balmohan V. Limaye

Department of Mathematics and Group of Theoretical Studies, Indian Institute of Technology, Bombay, India.