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ABSTRACT
Analysis tools for accessing to the general results of the spectral theory and the specific results concerning compact operators already exist, in particular those governing th etheory of nanlytical functions. However, the in-depth study of normal operators requires supplementary tools: the theory of measurement, topologies derived from a seminorm family as well the algebraic notion of ideal and the axiom of choice. this article presents several aspects of the spectral theorem of normal operators. Indeed, the Dunford integral allows for designing projectors which reduce the operator according to its elementary compounds. However, in the absence of the decomposition of the spectrum into related compounds, the design of projectors requires recoursing to measurement theory tools.
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Marc LENOIR: Research Director, CNRS - École nationale supérieure des techniques avancées
INTRODUCTION
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.
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The spectral theorem
Bibliography
Websites
PAULIN (F.). – Complements to spectral theory and harmonic analysis http://www.math.u-psud.fr/∼ paulin/notescours/cours_magistere2.pdf
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