Communications in Mathematical Physics

, Volume 164, Issue 3, pp 563–597 | Cite as

The Berezin transform and invariant differential operators

  • A. Unterberger
  • H. Upmeier


The Berezin calculus is important to quantum mechanics (creation-annihilation operators) and operator theory (Toeplitz operators). We study the basic Berezin transform (linking the contravariant and covariant symbol) for all bounded symmetric domains, and express it in terms of invariant differential operators.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Differential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B1] Berezin, F.A.: Quantization. Math. USSR Izvestija8, 1109–1165 (1974)Google Scholar
  2. [B2] Berezin, F.A.: Quantization in complex symmetric spaces. Math. USSR Izvestija9, 341–378 (1975)Google Scholar
  3. [B3] Berezin, F.A.: A connection between the co- and contravariant symbols of operators on classical complex symmetric spaces. Soviet Math. Dokl.19, 786–789 (1978)Google Scholar
  4. [BK] Braun, H., Koecher, M.: Jordan-Algebren. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  5. [BLU] Borthwick, D., Lesniewski, A., Upmeier, H.: Non-perturbative deformation quantization of Cartan domains. J. Funct. Anal.113, 153–176 (1993)Google Scholar
  6. [BBCZ] Békollé, D., Berger, C.A., Coburn, L.A., Zhu, K.H.: BMO in the Bergman metric on bounded symmetric domains. J. Funct. Anal.93, 310–350 (1990)Google Scholar
  7. [CGR] Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds II. Trans. Am. Math. Soc. (to appear)Google Scholar
  8. [F1] Faraut, J.: Jordan algebras and symmetric cones. Oxford University Press 1994Google Scholar
  9. [FK] Faraut, J., Korányi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal.88, 64–89 (1990)Google Scholar
  10. [H1] Helgason, S.: A duality in integral geometry on symmetric spaces. Proc. US-Japan Seminar Kyoto (1965), Tokyo: Nippon Hyronsha 1966Google Scholar
  11. [H2] Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962Google Scholar
  12. [H3] Helgason, S.: Groups and geometric analysis. New York: Academic Press 1984Google Scholar
  13. [K1] Karpelevic, F.I.: Orispherical radial parts of Laplace operators on symmetric spaces. Sov. Math.3, 528–531 (1962)Google Scholar
  14. [KS1] Korányi, A., Stein, E.:H 2 spaces of generalized half-planes. Studia Math.14, 379–388 (1972)Google Scholar
  15. [KS2] Kostant, B., Sahi, S.: The Capelli identity, tube domains, and the generalized Laplace transform. Adv. Math.87, 71–92 (1991)Google Scholar
  16. [KU] Kaup, W., Upmeier, H.: Jordan algebras and symmetric Siegel domains in Banach spaces. Math. Z.157, 179–200 (1977)Google Scholar
  17. [L1] Loos, O.: Bounded symmetric domains and Jordan pairs. Univ. of California, Irvine 1977Google Scholar
  18. [L2] Loos, O.: Jordan Paris. Lect. Notes in Math.460, Berlin: Springer-Verlag 1975Google Scholar
  19. [O1] Oersted, B.: A model for an interacting quantum field. J. Funct. Anal.36, 53–71 (1980)Google Scholar
  20. [P1] Perelomov, A.: Generalized coherent states and their applications. Berlin, Heidelberg, New York: Springer 1986Google Scholar
  21. [P2] Peetre, J.: The Berezin transform and Haplitz operators. J. Oper. Th.24, 165–186 (1990)Google Scholar
  22. [RV] Rossi, H., Vergne, M.: Analytic continuation of the holomorphic discrete series of a semisimple Lie group, Acta Math.136, 1–59 (1976)Google Scholar
  23. [S1] Schmid, W.: Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen. Invent. Math.9, 61–80 (1969)Google Scholar
  24. [U1] Upmeier, H.: ToeplitzC *-algebras on bounded symmetric domains. Ann. Math.119, 549–576 (1984)Google Scholar
  25. [U2] Unterberger, A.: Quantification de certains espaces hermitiens symétriques. Séminaire Goulaouic-Schwartz (1979–80), Ecole Polytechnique, ParisGoogle Scholar
  26. [U3] Upmeier, H.: JordanC *-algebras and symmetric Banach manifolds. Amsterdam: North-Holland 1985Google Scholar
  27. [U4] Upmeier, H.: Jordan algebras in analysis, operator theory and quantum mechanics. CBMS Conference Series 67 (1987)Google Scholar
  28. [U5] Upmeier, H.: Jordan algebras and harmonic analysis on symmetric spaces. Am. J. Math.108, 1–25 (1986)Google Scholar
  29. [U6] Upmeier, H.: Toeplitz operators on bounded symmetric domains. Trans. Am. Math. Soc.280, 221–237 (1983)Google Scholar
  30. [U7] Unterberger, A., Unterberger, J.: Quantification et analyse pseudo-différentielle. Ann. Scient. Éc. Norm. Sup.21, 133–158 (1988)Google Scholar
  31. [U8] Uribe, A.: A symbol calculus for a class of pseudodifferential operators onS n and band asymptotics. J. Funct. Anal.59, 535–556 (1984)Google Scholar
  32. [W] Wallach, N.: The analytic continuation of the discrete series. Trans. Am. Math. Soc.251, 1–37 (1979)Google Scholar
  33. [Y1] Yau, S.-T.: Uniformization of geometric structures. Proc. Symp. Pure Math.48, 265–274 (1988)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • A. Unterberger
    • 1
  • H. Upmeier
    • 2
  1. 1.Département de MathématiquesUniversité de ReimsReims CedexFrance
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA

Personalised recommendations