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Transactions of the American Mathematical Society

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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
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Abstract | References | Similar Articles | Additional Information


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.

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  • 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

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Additional Information

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

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