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Théories Spectrales: Chapitres 3 À 5
Paperback
ISBN13: 9783031195044
Publisher: Springer Nature
Published: Apr 11 2023
Pages: 566
Weight: 1.77
Height: 1.18 Width: 6.14 Depth: 9.21
Language: French
Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.
Ce second volume, inédit, du Livre consacré aux Théories spectrales a pour thème les propriétés spectrales des applications linéaires.
Le chapitre 3 étudie les applications linéaires compactes entre espaces vectoriels topologiques et la théorie de la perturbation par addition d'une application linéaire compacte, en particulier la théorie de Fredholm. Il se poursuit par la description du spectre d'un endomorphisme compact d'un espace de Banach, notamment les notions de spectre sensible et de spectre essentiel. On y démontre le théorème de Krein--Rutman.
Le chapitre 4 contient les résultats fondamentaux de la théorie spectrale hilbertienne: opérateurs compacts et nucléaires, endomorphismes normaux, opérateurs partiels normaux. On y trouve également un exposé concis des distributions et distributions tempérées.
Enfin, le chapitre 5 aborde l'étude des représentations unitaires des groupes topologiques (constructions élémentaires, lemme de Schur, représentations de carré intégrable modulo le centre, classes de représentations irréductibles). On y développe aussi la théorie des fonctions de type positif et on y démontre le théorème fondamental de Peter--Weyl.
Le texte est complété par de nombreux exercices et par une note historique portant sur le contenu des chapitres 1 à 5.
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