Abstract
In this chapter we address the following subjects: the spectrum and eigenvalues of an operator; the resolvent and non-emptyness of the spectrum; finite-rank operators, the approximation property and compactness in Banach spaces; compactness criteria for sets in specific spaces; definition and properties of compact operators; operators of the form \(I - T\) with T a compact operator; the structure of the spectrum of a compact operator.
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Notes
- 1.
Actually, there is an entire group of properties of “approximation property” type (see [28]). What we call here the pointwise approximation property should be more accurately referred to as the bounded approximation property for separable spaces.
- 2.
Here the symbol \(\otimes \) is read as tensor product.
- 3.
Clearly, it is unfortunate that we use the same letter K to denote the class of compact operators as well as the kernel of a kernel operator, not to speak of using it to denote compact spaces when we are dealing with the space C(K). But what can we do: these notations are widely accepted. To make the reader even happier, we could, as customary, also denote a compact operator by K! But enough is enough.
- 4.
The reader should note that here closedness in the sense of pointwise convergence does not hold.
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Kadets, V. (2018). Elements of Spectral Theory of Operators. Compact Operators. In: A Course in Functional Analysis and Measure Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-92004-7_11
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DOI: https://doi.org/10.1007/978-3-319-92004-7_11
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