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Eigenelements of perturbed operators

Part of: General theory of linear operators Approximations and expansions

Published online by Cambridge University Press:  09 April 2009

B. V. Limaye  and
M. T. Nair
B. V. Limaye
Affiliation:
Indian Institute of TechnologyPowai, Bombay 400076, India
M. T. Nair
Affiliation:
University of GoaSanta Cruz, Goa 403005, India
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Abstract

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Let λ0 be a semisimple eigenvalue of an operator T0. Let Γ0 be a circle with centre λs0 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 linear perturbation family T(t) = T0 + tV0 and the circle Γ0. They are useful when T0 is a member of a sequence (Tn) 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. I λ0 is simple, it is shown that the same lower bound are valid for the convergence radius of a power series yielding an eigenvector of T(t).

MSC classification

Secondary: 47A55: Perturbation theory 47A70: (Generalized) eigenfunction expansions; rigged Hilbert spaces 47A10: Spectrum, resolvent 41A35: Approximation by operators (in particular, by integral operators)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 1 , August 1990 , pp. 138 - 148
Copyright
Copyright © Australian Mathematical Society 1990

References

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