Journal of Operator Theory

Volume 35, Issue 2, Spring 1996 pp. 241-280.

C*-algebra techniques in numerical analysis

Authors: Steffen Roch (1) and Bernd Silbermann (2)
Author institution:(1) Universität Leipzig, Mathematisches Institut, Augustusplatz 10-11, D-04109 Leipzig, GERMANY
(2) Technische Universität Chemnitz-Zwickau, Fakultät für Mathematik, D-09107 Chemnitz, GERMANY

Summary: The topic of the present paper is a general approach of studying invertibility problems in algebras the elements of which are sequences of operators. These sequences can be viewed as approximation sequences for a given operator, and the proposed approach allows to relate properties of the approximation sequence (stable convergence, limiting sets of spectra, Moore-Penrose invertibility, asymptotic behaviour of the condition numbers) with corresponding properties of a certain function, the symbol of the sequence. This method applies to practically relevant approximation methods such as the finite section method for Toeplitz operators and spline projection methods for singular integral equations with piecewise continuous coefficients as well as for Mellin operators.

Keywords: Approximation methods, C*-algebra techniques, Toeplitz operators.

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