Semi-groups and Representations of Lie Groups

With every Lie semi-group, Π, possessing certain regularity properties, there is associated a Lie algebra, A; and with every strongly continuous representation of Π in a Banach space there is associated a representation A(a) of A. Certain theorems regarding this representation are established. The above theorems are valid for a representation of a Lie group also. In this case, it is shown that it is possible to extend the representation to elliptic elements of the universal enveloping algebra. It is also shown that the representatives of the strongly elliptic elements of the universal enveloping algebra are the infinitesimal generators of holomorphic semi-groups. Integral representations of these semigroups are given. † A dissertation presented to the Faculty of the Graduate School of Yale University in candidacy for the degree of Doctor of Philosophy, 1960 Semi-groups and representations of Lie groups 2 INTRODUCTION The study of Lie semi-groups and their representations was initiated by E. Hille in [6]. For a survey of the basic problems and results the reader is referred to that paper and to Chapter XXV of [7]. This thesis is a continuation of work begun there; we summarize briefly the results it contains. In Chapter I, the “Dense Graph Theorems” suggested in [6] are proved and it is shown that linear combinations of the infinitesimal generators form, in the precise sense of Theorems 4 and 6, a representation of a Lie algebra canonically associated with the semi-group. In Chapter II the study of the infinitesimal generators is continued. For the work of this chapter it is necessary to assume that the semi-group is a full Lie group. It is shown (Theorem 7) that the representation of the Lie algebra can be extended, in a natural manner, to a representation of the elliptic elements of the universal enveloping algebra. Then the spectral properties of operators corresponding to strongly elliptic elements are discussed; in particular it is shown (Theorem 8) that they are the infinitesimal generators of semi-groups holomorphic in a sector of the complex plane. Canonical representations of these semi-groups as integrals are given in Theorem 9. The reader interested in other work to which that of Chapter II is related is referred to [9], [13], [19], and a forthcoming paper by E. Nelson. Acknowledgement. The author wishes to thank C. T. Ionescu Tulcea for his advice and encouragement during the preparation of this dissertation. Semi-groups and representations of Lie groups 3