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

##### 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.)