J Austral Math Soc Ser A 49 pp138--148, 1990.

Eigenelements of Perturbed Operators

B. V. Limaye and M. T. Nair

Abstract

Let l₀ be a semisimple eigenvalue of an operator T₀. Let G₀ be a circle with centre l₀ containing no other spectral value of T₀. Some lower bounds are obtained for the convergence radius of the power series for the spectral projection P(t) (and for trace T(t)P(t)) associated with a linear perturbation family T(t) = T₀ + tV₀ and the circle G₀. They are useful when T₀ is a member os a sequence (T_n) which approximates an operator T in a collectively compact manner. These bounds result from a modification of Kato's method of majorizing series, based on an idea of Redont. If l₀ is simple, it is shown that the same lower bounds are valid for the convergence radius of a power series yielding an eigenvector of T(t).

1980 AMS Subject Classification (1985 Revision): 47A55, 47A70, 47A10, 41A35

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Authors

B. V. Limaye

Indian Institute of Technology, Powai, Bombay 400076, India.

M. T. Nair

Univeristy of Goa, Santa Cruz, Goa 403005, India.