Introduction

In Chapters III to V of this Book, we consider in detail the endomorphism algebra of a real or complex topological vector space. In the case of spaces of finite dimension, the principal properties of the elements of this algebra, taken individually, result from the Jordan decomposition (A, VII, p. 42, § 5, n$^o9$). We study here their generalization to spaces of infinite dimension, which is, however, more delicate.

In Chapter III, we consider, in a very general setting, compact linear mappings between topological vector spaces. These are those whose spectral properties come closest to the case of finite dimension. For example, in the particular case of Banach spaces, the non-zero elements of the spectrum of a compact endomorphism are generalized eigenvalues of finite spectral multiplicity.

In the context of Hilbert spaces, a classification and a manageable structure emerge. This is the object of Chapter IV. It begins with the study of compact operators in Hilbert spaces: their description by singular values, as well as their diagonalization in the case of a normal endomorphism, are quite analogous to the known properties for a space of finite dimension. More generally, the spectral theorem furnishes simple models for normal endomorphisms of Hilbert spaces.

Certain important applications, concerning for example differential operators, but also Quantum Mechanics, require a subtler theory: that of linear mappings, not necessarily continuous, defined on a dense subspace of a Hilbert space, which we call partial operators. In this setting, the

xi very definition of a self-adjoint or normal operator demands great precision if one wishes to avoid gross blunders. Once these definitions have been acquired, a classification similar to that of continuous endomorphisms emerges; the functional calculus for universally measurable mappings, bounded or not, furnishes close links between continuous normal endomorphisms and normal partial operators.

In passing, we shall have mentioned normal endomorphisms of trace, and in particular integral endomorphisms with kernel in a Hilbert space $L^2(X, \mu)$. The crucial importance is known, in the theory of automorphic forms and representations, of the formula which establishes that the trace of such an endomorphism, when it exists, is the integral of the kernel over the diagonal. We also devote a paragraph to the definitions and fundamental properties of distributions and tempered distributions in $\mathbf{R}^n$. These notions furnish not only the most convenient presentation of the fundamental principles of euclidean Fourier theory (cf. II, § 1), but they also make it possible to identify the domains of many important differential operators.

Finally, in Chapter V, we relate the study of endomorphisms of Hilbert spaces to that of unitary representations of groups. After some elementary general considerations, among which Schur’s Lemma is the most important statement, we study the case of locally compact groups. We then consider functions of positive type, and present their first applications. One of them is the Gelfand-Naimark-Segal theorem, which asserts that every star algebra is isomorphic to a closed subalgebra of the endomorphism algebra of a Hilbert space. Others concern fundamental properties of the space of irreducible unitary representations of a locally compact group, most particularly the Gelfand-Raikov theorem, which establishes that these representations separate the points. In the last paragraph of this chapter, we prove the fundamental results concerning representations of compact groups, which served us in Chapter IX of Groupes et algèbres de Lie.

In addition to the original articles and memoirs, we have benefited very particularly from reading the following works:

A. Borel, Représentations de groupes localement compacts, Lecture Notes in Mathematics 276, Springer, 1972.

M. Reed and B. Simon, Methods of Mathematical Physics, Academic Press, Vol. 1, 1980 and Vol. 2, 1975.

K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012.

In the preceding chapters and books, the following references to “TS, forthcoming” must be modified as follows:

EVT, V, p. 35, line 1. Instead of “TS, IV, § 6”, read “(TS, IV, p. 160,

remark)”.

INT, IX, p. 94, § 6, n$^o12$. Instead of “One can prove a converse . . .

bounded”, read “The converse of Prop. 11 is a particular case of TS, V,

p. 455, th. 5.”

LIE, IX, p. 61, § 6, n$^o4$, line 6. Instead of “TS, forthcoming”, read “TS, III,

p. 36, prop. 9”.

LIE, IX, p. 63, § 6, n$^o4$, line 6. Instead of “TS, forthcoming”, read “TS, III,

p. 36, rem.”.

LIE, IX, p. 66, § 7, note 1. Instead of “to a forthcoming chapter”, read “to

paragraph 4 of Chapter V”.

LIE, IX, p. 69, § 7, n$^o2$. Instead of “then according to TS”, read “then (TS, V,

p. 464, prop. 6)”.

LIE, IX, p. 71, § 7, n$^o3$. Instead of “which, according to TS, is an isomorphism”,

read “which, according to TS, V, p. 469, cor. 2, is an isomorphism.”

LIE, IX, p. 72, § 7, n$^o4$. Instead of “(TS)”, read “(TS, V, p. 459, cor. 3)”.

LIE, IX, p. 74, § 7, n$^o4$. After “of the space of central representative functions

on G”, add “(TS, V, p. 469, cor. 2, a))” ; after “of the space ZL$^2(G)$”

of square-integrable central functions on G », add « (TS, V, p. 468,

prop. 9). »

LIE, IX, p. 78, § 8, n$^o1$. Instead of « In this number, definitions

and results from TS are recalled », read « Definitions and results from TS are recalled, V,

p. 386, n$^o8$; p. 392, n$^o10$; p. 470, n$^o7$. »

LIE, IX, p. 84, § 8, n$^o3$. Instead of « Let us recall (TS) the formulae », read « One has

(TS, V, p. 467, formulae (1) and (2)) the formulae ».

LIE, IX, p. 89, § 9, n$^o1$. Instead of « TS, III, §2, n$^o7$, prop. 16 », read « TS, III,

p. 67, prop. 10 ».

LIE, IX, p. 90, § 9, n$^o1$. Instead of « TS, III, §2, n$^o6$ », read « TS, III, p. 61,

No. 4 ».

LIE, IX, p. 91, § 9, n$^o2$. Instead of « TS, to appear », read « TS, III, p. 36,

prop. 9 and TS, V, p. 464, prop. 7 ».

LIE, IX, p. 92, note 1. Instead of « (TS, to appear) », read « (TS, V, p. 379,

def. 3) ».

LIE, IX, p. 99, appendix 1, n$^o1$. Instead of « (TS, to appear) », read « (TS, V,

p. 464, remark 4) ».

LIE, IX, p. 128, exercise 3 (b) of §9. Instead of « (TS, to appear) », read « (Use

the Peter–Weyl theorem, TS, V, p. 462, th. 1) ».

TS, I, p. 103, example 6. Instead of « V, to appear », read « V, p. 442, th. 2. » TS, I, p. 173, exercise 5, d). Instead of « III, to appear », read « III, p. 82, § 6. » TS, II, p. 237, n$^o9$. Instead of « IV, to appear », read « IV, p. 196, § 3 ».

TS, II, p. 251, n$^o1$. Instead of « V, to appear », read « V, p. 425, prop. 10 ».