# The Spectrum of a Closed Operator

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## Abstract

Chapter 2 is about the spectrum and the resolvent of closed operators on Hilbert space. These are undoubtedly the most important general concepts in operator theory. In the first section, regular points and defect numbers of linear operators are defined and studied, and the Krasnoselskii–Krein theorem about the constancy of defect numbers on connected components of the regularity domain is proved. These results are used to derive basic properties of the spectrum and the resolvent of closed operators. Parts of the spectrum are discussed. The two resolvent identities, the spectral radius, and the analyticity of the resolvent are treated. Spectra and formulas for the resolvents of the differentiation operator \(-\mathrm{i} \frac{d}{dx}\) on various intervals are determined.

## Keywords

Hilbert Space Spectral Radius Closed Operator Regular Point Point Spectrum## References

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