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Mathematics of Computation

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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: Let $ M$ be a self-adjoint operator acting on a Hilbert space $ \mathcal{H}$. A complex number $ z$ is in the second order spectrum of $ M$ relative to a finite-dimensional subspace $ \mathcal{L}\subset \mathrm{Dom}\, M^2$ iff the truncation to $ \mathcal{L}$ of $ (M-z)^2$ 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 $ M,$ the uniform limit of these sets, as $ \mathcal{L}$ increases towards $ \mathcal{H}$, contain the isolated eigenvalues of $ M$ 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

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.