Abstract
This chapter describes the various elementary properties of the spectrum. We will first discuss the important case of bounded operators, and especially the remarkable resolvent bound for normal operators. Then, we will progressively consider more general closed operators and discuss the famous Riesz projections. Finally, we will say a few words about the Fredholm operators (and their indices). The main reason to do that is to define the discrete and the essential spectrum of a closed operator. Somehow, we will see that the Fredholm operators of index 0 are very close to being square matrices, at least from the spectral point of view.
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Notes
- 1.
Hint: use Theorem 6.2, which will be proved later.
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Cheverry, C., Raymond, N. (2021). SPECTRUM. In: A Guide to Spectral Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-67462-5_3
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DOI: https://doi.org/10.1007/978-3-030-67462-5_3
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Publisher Name: Birkhäuser, Cham
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