Analysis on symmetric cones / Jacques Faraut and Adam Koranyi

The contents of the book are the following: In the first chapter symmetric cones are introduced, and in the second chapter the basic properties of Jordan algebras are presented. With the aid of these in Chapter III the fundamental result due to M. Koecher and E. B. Vinberg can be stated that the interior of the cone of squares in a Euclidean Jordan algebra is a symmetric cone, and every symmetric cone is obtained in this way. In Chapter IV the Peirce decomposition is presented. This gives an algebraic technique making it possible to regard a Jordan algebra as a generalization of the space of symmetric matrices. It allows us to give a classification of Euclidean Jordan algebras and symmetric cones in Chapter V. In Chapter VI generalized versions of the diagonalization and triangularization of matrices are discussed. The interplay of Euclidean harmonic analysis and of non-commutative harmonic analysis gives rise to the theory of the gamma function of a symmetric cone which is introduced in Chapter VII. It plays a central role in this book. In Chapter VIII it is proved that any semisimple Jordan algebra is the complexification of a Euclidean one.

In Chapter IX complex tube domains over convex cones are studied and the discussion of the Bergman and Hardy spaces on the tube domain is given. In Chapter X tube domains over symmetric cones are specialized and it allows us to obtain explicit formulas for the Bergman and Cauchy-Szegö kernels involving the determinant function of the underlying Jordan algebra. Another point where Euclidean analysis and non-commutative harmonic analysis come together is the theory of spherical polynomials, introduced in Chapter XI. They can be used to define generalized Taylor and Laurent expansions which are studied in Chapter XII.

In Chapter XIII the function spaces on symmetric domains of tube type are studied and the Wallach set is determined. In Chapter XIV it is shown that the spherical functions of a symmetric cone, considered as a Riemannian symmetric space, and the spherical Fourier transform, can be written explicitly in terms of the Jordan algebra structure. The gamma function of a symmetric cone leads naturally to generalized hypergeometric expansions which are studied in Chapter XV, with particular emphasis on the Bessel functions and the Gauss hypergeometric functions. In Chapter XVI it is shown that in some cases it is possible to consider a symmetric cone as a set of dilations of a representation space of the Jordan algebra and to obtain in that way analogs of some classical objects such as the Hankel transform. (Zentralblatt)

Bibliogr. p. 364-377. Index