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Pré-Publication, Document De Travail Année : 2010

Weyl theorems for the polluted set of self-adjoint operators in Galerkin approximations

Lyonell Boulton
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Mathieu Lewin

Résumé

Let A be a self-adjoint operator on a separable Hilbert space H and let (L_n) be a sequence of finite dimensional subspaces of the domain of A, approximating H in the large n limit. Denote by A_n the compression of A to L_n. In general the spectrum of A is only a subset of the limit of the spectra of A_n and the latter might differ from the former in a non-trivial ``polluted set''. In this paper we show that this polluted set is determined by the existence of particular Weyl sequences of singular type. This characterization allows us to identify verifiable conditions on self-adjoint perturbations B, ensuring that the polluted set of B is identical to that of A. The results reported are illustrated by means of several canonical examples and they reveal the many subtleties involved in the systematic study of spectral pollution.
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Dates et versions

hal-00536270 , version 1 (15-11-2010)
hal-00536270 , version 2 (22-11-2011)

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Lyonell Boulton, Nabile Boussaid, Mathieu Lewin. Weyl theorems for the polluted set of self-adjoint operators in Galerkin approximations. 2010. ⟨hal-00536270v1⟩
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