Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
Published electronically: May 4, 2001
MathSciNet review: 1837258
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

Let $\mathbb D$ be a bounded symmetric domain in a complex vector space $V_{\mathbb C}$with a real form $V$ and $D=\mathbb D\cap V=G/K$ be the real bounded symmetric domain in the real vector space $V$. We construct the Berezin kernel and consider the Berezin transform on the $L^2$-space on $D$. The corresponding representation of $G$is then unitarily equivalent to the restriction to $G$of a scalar holomorphic discrete series of holomorphic functions on $\mathbb D$ and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the $L^2$-space.


References [Enhancements On Off] (What's this?)

  • 1. Jonathan Arazy and Gen Kai Zhang, 𝐿^{𝑞}-estimates of spherical functions and an invariant mean-value property, Integral Equations Operator Theory 23 (1995), no. 2, 123–144. MR 1351341, 10.1007/BF01197533
  • 2. F. A. Berezin, General concept of quantization, Comm. Math. Phys. 40 (1975), 153–174. MR 0411452
  • 3. Jacques Faraut and Adam Korányi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1446489
  • 4. Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
  • 5. Joachim Hilgert and Gestur Ólafsson, Causal symmetric spaces, Perspectives in Mathematics, vol. 18, Academic Press, Inc., San Diego, CA, 1997. Geometry and harmonic analysis. MR 1407033
  • 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, Translated from the Russian by Leo Ebner and Adam Korányi, American Mathematical Society, Providence, R.I., 1963. MR 0171936
  • 8. Bertram Kostant and Siddhartha Sahi, Jordan algebras and Capelli identities, Invent. Math. 112 (1993), no. 3, 657–664. MR 1218328, 10.1007/BF01232451
  • 9. O. Loos, Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977.
  • 10. Yu. Neretin, Matrix analogs of the integral $B(\alpha, \rho-\alpha)$ and Plancherel formula for Berezin kernel representations, (1999), preprint, Math.RT/9905045.
  • 11. G. Ólafsson, Causal symmetric spaces, Mathematica Gottingensis 15 (1990).
  • 12. Gestur Ólafsson and Bent Ørsted, Generalizations of the Bargmann transform, Lie theory and its applications in physics (Clausthal, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 3–14. MR 1634480
  • 13. Bent Ørsted and Gen Kai Zhang, Weyl quantization and tensor products of Fock and Bergman spaces, Indiana Univ. Math. J. 43 (1994), no. 2, 551–583. MR 1291529, 10.1512/iumj.1994.43.43023
  • 14. Bent Ørsted and Genkai Zhang, 𝐿²-versions of the Howe correspondence. I, Math. Scand. 80 (1997), no. 1, 125–160. MR 1466908
  • 15. Jaak Peetre, The Berezin transform and Ha-plitz operators, J. Operator Theory 24 (1990), no. 1, 165–186. MR 1086552
  • 16. Goro Shimura, Generalized Bessel functions on symmetric spaces, J. Reine Angew. Math. 509 (1999), 35–66. MR 1679166, 10.1515/crll.1999.041
  • 17. A. Unterberger and H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), no. 3, 563–597. MR 1291245
  • 18. G. van Dijk and S. C. Hille, Canonical representations related to hyperbolic spaces, J. Funct. Anal. 147 (1997), no. 1, 109–139. MR 1453178, 10.1006/jfan.1996.3057
  • 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 $SL(2, \mathbf R)$ where $\mathbf R$ is a ring of functions, Uspekhi Mat. Nauk 28 (1973), no. 5, 83-128.
  • 21. Genkai Zhang, Berezin transform on line bundles over bounded symmetric domains, J. Lie Theory 10 (2000), no. 1, 111–126. MR 1748086

Similar Articles

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: http://dx.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