C*-algebras and Asymptotic Spectral Theory
The presented material is a slightly polished and extended version of lectures given at Lisbon, WOAT 2006. Three basic topics of numerical functional analysis are discussed: stability, fractality, and Fredholmness. It is further shown that these notions are corner stones in order to understand a few topics in asymptotic spectral theory: asymptotic behavior of singular values, ε-pseudospectra, norms. Four important examples are discussed: Finite sections of quasidiagonal operators, Toeplitz operators, band-dominated operators with almost periodic coefficients, and general band-dominated operators. The elementary theory of C*-algebras serves as the natural background of these topics.
KeywordsC*-algebras operator sequences asymptotics finite sections
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