A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains. (English) Zbl 0831.46014

Arazy, Jonathan

Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 7-65 (1995).

This paper is a survey of recent developments concerning Hilbert spaces \(H\) of analytic functions on bounded symmetric domains \(D\) which are invariant under biholomorphic automorphisms of \(D\), as well as the representation theory of these automorphism groups. The survey is written for a general audience of functional analysts, and some of the works cited are in preprint form.
After reviewing E. Cartan’s classification of \(D\) and the Jordan triple product, the author presents various ways of describing \(H\), including the work of J. Faraut and A. Kornyi [J. Funct. Anal. 88, No. 1, 64-89 (1990; Zbl 0718.32026)] and joint work of the author with S. D. Fisher [Oper. Theory: Adv. Appl. 48, 67-91 (1990; Zbl 0733.46011)], as well as extensions by the author of work by Z. Yan to invariant inner products for Cartan tube domains. He illustrates some of the results by formulating them in case \(D\) is a classical Cartan domain regarded as a space of complex matrices or as the unit ball in complex \(n\)-space, where formulas for the inner products on \(H\) are derived. There are many highly technical results drawing upon various fields of both classical and functional analysis.
For the entire collection see [Zbl 0819.00022].

Reviewer: J.V.Whittaker (Vancouver)

MSC:

46E20	Hilbert spaces of continuous, differentiable or analytic functions
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
43A85	Harmonic analysis on homogeneous spaces
46H70	Nonassociative topological algebras
32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)

Keywords:

Bergman kernel; Peter-Weyl decomposition; Hilbert spaces of analytic functions on bounded symmetric domains; invariant under biholomorphic automorphisms; Jordan triple product; invariant inner products for Cartan tube domains