Limiting set of second order spectra

Author:
Lyonell Boulton

Journal:
Math. Comp. **75** (2006), 1367-1382

MSC (2000):
Primary 47B36; Secondary 47B39, 81-08

Published electronically:
February 22, 2006

MathSciNet review:
2219033

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a self-adjoint operator acting on a Hilbert space . A complex number is in the second order spectrum of relative to a finite-dimensional subspace iff the truncation to of is not invertible. This definition was first introduced in Davies, 1998, and according to the results of Levin and Shargorodsky in 2004, these sets provide a method for estimating eigenvalues free from the problems of spectral pollution. In this paper we investigate various aspects related to the issue of approximation using second order spectra. Our main result shows that under fairly mild hypothesis on the uniform limit of these sets, as increases towards , contain the isolated eigenvalues of of finite multiplicity. Therefore, unlike the majority of the standard methods, second order spectra combine nonpollution and approximation at a very high level of generality.

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Additional Information

**Lyonell Boulton**

Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, United Kingdom

Email:
lyonell@ma.hw.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-06-01830-8

Keywords:
Second order spectrum,
projection methods,
spectral pollution,
numerical approximation of the spectrum.

Received by editor(s):
July 16, 2003

Received by editor(s) in revised form:
April 18, 2005

Published electronically:
February 22, 2006

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.