Abstract
By definition, two operators S, T ∈ L (H) have the same image in the Calkin algebra \(Q\;(H)\;: = \;L(H)\;/\;K(H)\) if and only if S is a compact perturbationof T, i.e., S = T + K for some compact operator K As we shall see later on, the compact operators can be regarded as “infinitesimal elements” of ℒ(ℋ), and it is of interest to know what properties of an operator are unchanged by compact perturbations. For instance, an invertible operator does not remain invertible (think of1–|ξ > <ξ| where ξ ∈ ℋis any nonzero vector), but we may recall the following well–known result, called Atkinson’s theorem [367, Prop. 3.3.11].
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© 2001 Springer Science+Business Media New York
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Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H. (2001). Fredholm Operators on C*-modules. In: Elements of Noncommutative Geometry. Birkhäuser Advanced Texts. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0005-5_4
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DOI: https://doi.org/10.1007/978-1-4612-0005-5_4
Publisher Name: Birkhäuser, Boston, MA
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