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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Limiting set of second order spectra
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by Lyonell Boulton PDF
Math. Comp. 75 (2006), 1367-1382 Request permission

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 $\mathcal {L}\subset \operatorname {Dom} M^2$ 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.
References
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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
  • Received by editor(s): July 16, 2003
  • Received by editor(s) in revised form: April 18, 2005
  • Published electronically: February 22, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 75 (2006), 1367-1382
  • MSC (2000): Primary 47B36; Secondary 47B39, 81-08
  • DOI: https://doi.org/10.1090/S0025-5718-06-01830-8
  • MathSciNet review: 2219033