Mathematische Zeitschrift

, Volume 268, Issue 3–4, pp 931–967 | Cite as

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

  • Miroslav Engliš
  • Harald UpmeierEmail author


An analogue of the star product, familiar from deformation quantization, is studied in the setting of real bounded symmetric domains. The analogue turns out to be a certain invariant operator, which one might call star restriction, from functions on the hermitification of the domain into functions on the domain itself. In particular, we establish the usual (i.e. semiclassical) asymptotic expansion of this star restriction, and further obtain a real- variable analogue of a theorem of Arazy and Ørsted concerning the analogous expansion for the Berezin transform.


Bounded symmetric domain Toeplitz operator Star product Covariant quantization 

Mathematics Subject Classification (2000)

Primary 32M15 Secondary 46E22 47B35 53D55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arazy J., Ørsted B.: Asymptotic expansions of Berezin transforms. Indiana Univ. Math. J. 49, 7–30 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arazy J., Upmeier H.: Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains. In: Kufner, A., Cwikel, M., Engliš, M., Persson, L.-E., Sparr, G. (eds) Function Spaces, Interpolation Theory, and Related Topics (Lund, 2000), pp. 151–211. Walter de Gruyter, Berlin (2002)Google Scholar
  3. 3.
    Arazy, J., Upmeier, H.: Covariant symbolic calculi on real symmetric domains. Singular integral operators, factorization and applications. In: Operator Theory Advances and Applications, vol. 142, pp. 1–27. Birkhäuser, Basel (2003)Google Scholar
  4. 4.
    Arazy J., Upmeier H.: A one-parameter calculus for symmetric domains. Math. Nachr. 280, 939–961 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Berezin F.A.: Quantization. Math. USSR Izv. 8, 1109–1163 (1974)zbMATHCrossRefGoogle Scholar
  6. 6.
    Borthwick D., Lesniewski A., Upmeier H.: Non-perturbative deformation quantization on Cartan domains. J. Funct. Anal. 113, 153–176 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    van Dijk G., Pevzner M.: Berezin kernels of tube domains. J. Funct. Anal. 181, 189–208 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Engliš M.: Berezin–Toeplitz quantization and invariant symbolic calculi. Lett. Math. Phys. 65, 59–74 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Engliš M.: Berezin–Toeplitz quantization on the Schwartz space of bounded symmetric domains. J. Lie Theory 15, 27–50 (2005)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Engliš M.: Toeplitz operators and group representations. J. Fourier Anal. Appl. 13, 243–265 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Engliš, M., Upmeier, H.: Toeplitz quantization and asymptotic expansions: geometric construction. In: SIGMA Symmetry Integrability Geometry: Methods and Applications, vol. 5, Paper 021, p. 30 (2009)Google Scholar
  12. 12.
    Engliš, M., Upmeier, H.: Toeplitz quantization and asymptotic expansions: Peter–Weyl components (2010, forthcoming)Google Scholar
  13. 13.
    Faraut J., Korányi A.: Analysis on Symmetric Cones. Clarendon Press, Oxford (1994)zbMATHGoogle Scholar
  14. 14.
    Folland, G.B.: Harmonic analysis in phase space. In: Annals of Mathematical Studies, vol. 122. Princeton University Press, Princeton (1989)Google Scholar
  15. 15.
    Gangolli R., Varadarajan V.S.: Harmonic Analysis of Spherical Functions on Real Reductive Groups. Springer, New York (1985)Google Scholar
  16. 16.
    Helgason S.: Differential Geometry and Symmetric Spaces. Academic Press, New York (1962)zbMATHGoogle Scholar
  17. 17.
    Hörmander L.: The analysis of linear partial differential operators, vol. I. In: Grundlehren der mathematischen Wissenschaften, vol. 256. Springer, Berlin (1985)Google Scholar
  18. 18.
    Kaup W.: On real Cartan factors. Manuscr. Math. 92, 191–222 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Loos O.: Bounded Symmetric Domains and Jordan Pairs. University of California, Irvine (1977)Google Scholar
  20. 20.
    Loos, O.: Jordan pairs. Lecture Notes in Mathematics, vol. 460. Springer, Berlin (1975)Google Scholar
  21. 21.
    MacDonald I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Clarendon Press, Oxford (1995)zbMATHGoogle Scholar
  22. 22.
    Neretin Y.: Plancherel formula for Berezin deformation on L 2 on Riemann symmetric space. J. Funct. Anal. 189, 336–408 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Olafsson, G., Ørsted, B.: Generalizations of the Bargmann transform. In: Lie theory and its Applications in Physics (Clausthal 1995), pp. 3–14. World Scientific, River Edge (1996)Google Scholar
  24. 24.
    Ørsted B., Zhang G.: Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ. Math. J. 43, 551–582 (1994)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Unterberger A., Unterberger J.: La série discrète de SL(2, R) et les opérateurs pseudo-différentiels sur une demi-droite. Ann. Sci. École Norm. Sup. (4) 17, 83–116 (1984)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Unterberger A., Upmeier H.: Berezin transform and invariant differential operators. Comm. Math. Phys. 164, 563–598 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Upmeier H.: Toeplitz operators on bounded symmetric domains. Trans. Am. Math. Soc. 280, 221–237 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Zhang G.: Berezin transform on real bounded symmetric domains. Trans. Am. Math. Soc. 353, 3769–3787 (2001)zbMATHCrossRefGoogle Scholar
  29. 29.
    Zhang G.: Branching coefficients of holomorphic representations and Segal–Bargmann transform. J. Funct. Anal. 195, 306–349 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Zhang, G.: Degenerate principal series representations and their holomorphic extensions (2007, preprint). arXiv:math/0711.1480Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematics InstituteSilesian University at OpavaOpavaCzech Republic
  2. 2.Mathematics InstitutePrague 1Czech Republic
  3. 3.Fachbereich MathematikUniversität MarburgMarburgGermany

Personalised recommendations