Berezin transform on real bounded symmetric domains

Author:
Genkai Zhang

Journal:
Trans. Amer. Math. Soc. **353** (2001), 3769-3787

MSC (2000):
Primary 22E46, 43A85, 32M15, 53C35

DOI:
https://doi.org/10.1090/S0002-9947-01-02832-X

Published electronically:
May 4, 2001

MathSciNet review:
1837258

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be a bounded symmetric domain in a complex vector space with a real form and be the real bounded symmetric domain in the real vector space . We construct the Berezin kernel and consider the Berezin transform on the -space on . The corresponding representation of is then unitarily equivalent to the restriction to of a scalar holomorphic discrete series of holomorphic functions on and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the -space.

**1.**J. Arazy and G. Zhang,*-estimates of spherical functions and mean-value property*, Integral Equations and Operator Theory,**23**(1995), 123-144. MR**96g:22015****2.**F. A. Berezin,*General concept of quantization*, Commun. Math. Phys.**40**(1975), 153-174. MR**53:15186****3.**J. Faraut and A. Koranyi,*Analysis on symmetric cones*, Oxford University Press, Oxford, 1994. MR**98g:17031****4.**S. Helgason,*Differential geometry and symmetric spaces*, Academic Press, New York, London, 1978. MR**80k:53081****5.**J. Hilgert and G. Ólafsson,*Causal symmetric spaces, geometry and harmonic analysis*, Perspectives in Mathematics, vol. 18, Academic Press, 1997. MR**97m:43006****6.**S. C. Hille,*Canonical representations*, Ph.D. thesis, Leiden University, 1999.**7.**L. K. Hua,*Harmonic analysis of functions of several complex variables in the classical domains*, Amer. Math. Soc., Providence, Rhode Island, 1963. MR**30:2162****8.**B. Kostant and S. Sahi,*Jordan algebras and Capelli identities*, Invent. Math.**112**(1993), 657-664. MR**94b:17054****9.**O. Loos,*Bounded symmetric domains and Jordan pairs*, University of California, Irvine, 1977.**10.**Yu. Neretin,*Matrix analogs of the integral and Plancherel formula for Berezin kernel representations*, (1999), preprint, Math.RT/9905045.**11.**G. Ólafsson,*Causal symmetric spaces*, Mathematica Gottingensis**15**(1990).**12.**G. Ólafsson and B. Ørsted,*Generalizations of the Bargmann transform*, Lie theory and its applications in physics. Proceedings of the international workshop, Clausthal, Germany, August 14-17, 1995. (H.-D.Doebner et al, ed.), World Scientific, Singapore, 1996, pp. 3-14. MR**99e:22032****13.**B. Ørsted and G. Zhang,*Weyl quantization and tensor products of Fock and Bergman spaces*, Indiana Univ. Math. J.**43**(1994), 551-582. MR**95h:22008****14.**-,*-versions of the Howe correspondence 1*, Math. Scand.**80**(1997), 125-160. MR**99c:22017****15.**J. Peetre,*Berezin transform and Ha-plitz operators*, J. Oper. Theory**24**(1990), 165-168. MR**91k:47058****16.**G. Shimura,*Generalized Bessel functions on symmetric spaces*, J. Reine Angew. Math.**509**(1999), 35-66. MR**2000e:33020****17.**A. Unterberger and H. Upmeier,*The Berezin transform and invariant differential operators*, Comm. Math. Phys.**164**(1994), 563-597. MR**96h:58170****18.**G. van Dijk and S. C. Hille,*Canonical representations related to hyperbolic spaces*, J. Funct. Anal.**147**(1997), 109-139. MR**98k:22053****19.**G. van Dijk and M. Pevzner,*Berezin kernels and tube domains*, J. Funct. Anal., to appear.**20.**A. M. Vershik, I.M. Gel'fand, and M.I. Graev,*Representations of the group where is a ring of functions*, Uspekhi Mat. Nauk**28**(1973), no. 5, 83-128.**21.**G. Zhang,*Berezin transform on line bundles over bounded symmetric domains*, J. Lie Theory**10**(2000), 111-126. MR**2001c:32015**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
22E46,
43A85,
32M15,
53C35

Retrieve articles in all journals with MSC (2000): 22E46, 43A85, 32M15, 53C35

Additional Information

**Genkai Zhang**

Affiliation:
Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden

Email:
genkai@math.chalmers.se

DOI:
https://doi.org/10.1090/S0002-9947-01-02832-X

Keywords:
Real bounded symmetric domains,
Jordan triples,
Siegel domains,
Berezin transform,
invariant differential operators,
unitary representations of Lie groups,
irreducible decomposition

Received by editor(s):
January 16, 2000

Received by editor(s) in revised form:
October 10, 2000

Published electronically:
May 4, 2001

Additional Notes:
Research supported by the Swedish Natural Sciences Research Council (NFR)

Article copyright:
© Copyright 2001
American Mathematical Society