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Journal of Operator Theory

Volume 48, Issue 1, Summer 2002  pp. 121-149.

Algebras of approximation sequences: Fredholmness

Authors:  Steffen Roch
Author institution: Technische Universitaet Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany

Summary:  In this paper, a Fredholm theory for approximation sequences is proposed. A sequence is called Fredholm if it is invertible modulo a certain ideal which plays the role of the ideal of the compact operators in the Fredholm theory of operators. With every Fredholm sequence, there are associated three integers which are the analogues of the nullity, the deficiency and the index of a Fredholm operator. The nullity of a Fredholm sequence $(A_n)$ is interpreted as a quantity which describes the asymptotic behaviour of the small singular values of the matrices $A_n$ as $n \to \infty$, and an identity is derived which allows the computation of this nullity in many situations. Several examples and applications are discussed.

Keywords:  Fredholm theory, approximation methods, asymptotic behaviour of singular values

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