Abstract
This chapter recalls various elementary facts about compact operators. We prove the fundamental fact that \(K-z\mathrm {Id}\) is a Fredholm operator when \(z\ne 0\) and when \(K\in \mathcal {L}(E)\) is compact. This fact has important spectral consequences for compact operators (especially once we will have proved that the index of \(K-z\mathrm {Id}\) is actually 0).
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- 1.
To do so, consider \((T_n)\) defined by
$$\begin{aligned} T_n(\ell )=\ell (x_n)\,,\quad \forall \ell \in E'\,, \end{aligned}$$and notice that \(\Vert T_n\Vert \leqslant \Vert x_n\Vert \leqslant 1\).
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Cheverry, C., Raymond, N. (2021). COMPACT OPERATORS. In: A Guide to Spectral Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-67462-5_4
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DOI: https://doi.org/10.1007/978-3-030-67462-5_4
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-67461-8
Online ISBN: 978-3-030-67462-5
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