The spectral theorem

The analytical tools of analytic function theory and the theory of Banach and Hilbert spaces provide access to the general results of spectral theory and to specific results relating to compact operators. In-depth analysis of normal operators, i.e. those that commute with their adjoint and do not satisfy the compactness hypothesis, requires additional tools of various kinds: measure theory, topologies derived from a family of semi-norms, as well as the algebraic notion of ideal and the axiom of choice.

This document can be seen as a continuation of the article [AF 567] spectral theory and applications; its aim is to present various aspects of the spectral theorem for normal operators. When the spectrum is resolved into related components, and especially when it is discrete, Dunford's integral makes it possible to construct projectors reducing the operator according to its elementary components. This strategy remains valid in principle for the analysis of normal operators, but in the absence of a decomposition of the spectrum into related components, the construction of projectors requires recourse to the tools of measure theory.