Abstract
A basic idea to study the stability of an approximation sequence is to translate the stability problem into an invertibility problem in a suitably defined C*-algebra, which offers the applicability of C*-tools to problems in numerical analysis. This approach associates with every classGof operators and with every discretization procedureDfor the operators inGa concrete C*-algebra 21(GD).The surprising result of an analysis of these algebras is that, for many apparently quite different choices ofLandDthey show a common structure, which might be formally summarized in the so-called standard model.
The goal of this paper is to analyse how this formal structure arises from intrinsic properties of the algebras 2f. The basic of these properties are the fractality of the algebra and a certain maximality condition for an ideal in 2( which is intimately related to the Fredholm theory in 21. Here fractality of 2f means that every sequence in 2( can be completely reconstructed from each of its subsequences modulo a sequence tending to zero in the norm, whereas the Fredholm property of a sequence (An) is closely related to the asymptotic behavior of the smallest singular values of the matrices An.
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Roch, S. (2003). Algebras of Approximation Sequences: Structure of Fractal Algebras. In: Böttcher, A., Kaashoek, M.A., Lebre, A.B., dos Santos, A.F., Speck, FO. (eds) Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, vol 142. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8007-7_16
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DOI: https://doi.org/10.1007/978-3-0348-8007-7_16
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