By N. Bourbaki
Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.
Ce most appropriate quantity du Livre consacré aux Théorie spectrales, dernier Livre du traité, comprend les chapitres: -1. Algèbres normée; -2. Groupes localement compacts commutatifs.
Le most effective chapitre introduit des recommendations de base en examine fonctionnelle. Le deuxième chapitre a pour aboutissement los angeles transformation de Fourier sur les groupes localement compacts commutatifs.
Ce quantity est une réimpression de l’édition de 1967.
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Additional info for Théories spectrales : Chapitres 1-2
Sample text
For / G D m ) - {/ G AC[0,1] | / ' G Lp(0,1), /(0) - /(l)} 24 CHAPTER 1. LINEAR OPERATORS IN BANACH SPACES It can be easily seen that A is an eigenvalue of Ti iff eA = 1 iff A = 2inir, n = 0, ± 1 , . . The corresponding eigenvectors are eXx. 1. Thus a(Ti) =
Thus, we may assume that k = n. Define a compact S G 93(X) by 771 Sx = Tx-^2fi{x)yi. 2=1 If Sx = Xx, then for 1 < j < m we have 0 = gj{\x-Sx) = \gj(x)-9j(Tx) + fj(x) = A f t - W - ^ - J W + ^ i ) = /,(*). Therefore Sx = Tx = Xx and, also, ^j(x) = 0. Thus, x = 0 and £ — A is one-to-one. 12 implies that Sx — Xx = y m + i for some x, giving the contradiction that 1 = £m+i(ym+i) = £m+i(Tx) - Xgm+i{x) = (T*0 m+ i)(a;) - Ap m+ i(:r) = 0. 7. 12 are p e r h a p s t h e most often quoted properties of compact operators.
A linear operator T in a Banach space X is said to have c o m p a c t r e s o l v e n t if there exists Ao in t h e resolvent set of T such t h a t (T — A o ) - 1 is compact. 4. 6). Consider T 2 : D(T 2 ) C LP(R) -> L P (E), p G [1, oo], given by T 2 / = / ' , D(T 2 ) = {/ G L p (E) | / G AC[-a,a] for all a G (0,oo), / ' G L P (M)}. 6, it is shown that a(T2) = iE. 16 implies that it cannot have compact resolvent. We shall see that differential operators typically have compact resolvents when defined in bounded domains; when defined in unbounded domains, they usually do not have compact resolvents.
