Berezin transform on real bounded symmetric domains

Zhang, Genkai

doi:10.1090/S0002-9947-01-02832-X

Berezin transform on real bounded symmetric domains
HTML articles powered by AMS MathViewer

by Genkai Zhang PDF

Trans. Amer. Math. Soc. 353 (2001), 3769-3787 Request permission

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

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, DOI 10.1007/BF01197533
F. A. Berezin, General concept of quantization, Comm. Math. Phys. 40 (1975), 153–174. MR 411452
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
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
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
S. C. Hille, Canonical representations, Ph.D. thesis, Leiden University, 1999.
L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR 0171936
Bertram Kostant and Siddhartha Sahi, Jordan algebras and Capelli identities, Invent. Math. 112 (1993), no. 3, 657–664. MR 1218328, DOI 10.1007/BF01232451
O. Loos, Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977.
Yu. Neretin, Matrix analogs of the integral $B(\alpha , \rho -\alpha )$ and Plancherel formula for Berezin kernel representations, (1999), preprint, Math.RT/9905045.
G. Ólafsson, Causal symmetric spaces, Mathematica Gottingensis 15 (1990).
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
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, DOI 10.1512/iumj.1994.43.43023
Bent Ørsted and Genkai Zhang, $L^2$-versions of the Howe correspondence. I, Math. Scand. 80 (1997), no. 1, 125–160. MR 1466908, DOI 10.7146/math.scand.a-12615
Jaak Peetre, The Berezin transform and Ha-plitz operators, J. Operator Theory 24 (1990), no. 1, 165–186. MR 1086552
Goro Shimura, Generalized Bessel functions on symmetric spaces, J. Reine Angew. Math. 509 (1999), 35–66. MR 1679166, DOI 10.1515/crll.1999.041
A. Unterberger and H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), no. 3, 563–597. MR 1291245
G. van Dijk and S. C. Hille, Canonical representations related to hyperbolic spaces, J. Funct. Anal. 147 (1997), no. 1, 109–139. MR 1453178, DOI 10.1006/jfan.1996.3057
G. van Dijk and M. Pevzner, Berezin kernels and tube domains, J. Funct. Anal., to appear.
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.
Genkai Zhang, Berezin transform on line bundles over bounded symmetric domains, J. Lie Theory 10 (2000), no. 1, 111–126. MR 1748086