Function Spaces

  • Adam Korányi
Part of the Progress in Mathematics book series (PM, volume 185)


The interest of the Hilbert spaces L λ 2 , of Chapter V is due to the fact that the group G of holomorphic automorphisms of D acts on them by (irreducible) unitary representations. This statement must be made a little more precise. A unitary (resp. bounded) representation is properly speaking a strongly continuous homomorphism gT (g)into the unitary (resp. bounded) transformations of a Hilbert space, such that T (g) T (h) = T (gh) for all g,h∈G. A slightly more general notion is that of a (unitary, or bounded) projective representation,where one requires only T (g)T (h) = c (g, h)T (gh) with some c (g, h) ∈ ℂ.


Wallach Bonami 


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  1. [A]
    J. Arazy, Realization of the invariant inner products on the highest quotients of the composition series, Ark. Mat. 30(1992), 1–24.MathSciNetMATHCrossRefGoogle Scholar
  2. [AF]
    J. Arazy and S. Fisher, Invariant Hilbert spaces of analytic functions on bounded symmetric domains, Topics in Operator Theory: Ernst D. Hellinger memorial volume, 67–91, Operator Theory Adv. Appl. 48, Birkhäuser, Basel, 1990.Google Scholar
  3. [AFP]
    J. Arazy, S. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145.MathSciNetMATHGoogle Scholar
  4. [BB]
    D. Békollé and A. Bonami, Estimates for the Bergman and Szegö projections in two symmetric domains of Cn, Colloq. Math. 68 (1995), 81–100.MathSciNetMATHGoogle Scholar
  5. [BBCZ]
    D. Békollé, C. Berger, L. Coburn and K. Zhu, BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93(1990), 310–350.MathSciNetMATHCrossRefGoogle Scholar
  6. [FK1]
    J. Faraut and A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88(1990), 64–89.MathSciNetMATHCrossRefGoogle Scholar
  7. [FK2]
    J. Faraut and A. Korányi, Analysis on symmetric cones, Oxford University Press, Oxford, 1994.MATHGoogle Scholar
  8. [G]
    S.G. Gindikin, Analysis in homogeneous domains. Uspekhi Mat. Nauk 19, 4(1964), 3–92 (in Russian). English translation; Russian Math. Surveys 19, 4(1964), 1–90.MathSciNetMATHGoogle Scholar
  9. [HCh]
    Harish-Chandra, Representations of semisimple Lie groups, VI, Amer. J. Math. 78 (1956), 564–628.MathSciNetMATHCrossRefGoogle Scholar
  10. [111]
    S. Helgason, Differential Geometry, Lie Groups, and Sym-metric Spaces, Academic Press, New York, 1978.Google Scholar
  11. [H2]
    S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984.MATHGoogle Scholar
  12. [H3]
    S. Helgason, Geometric analysis on symmetric spaces, Amer. Math. Soc., Providence, RI, 1994.MATHGoogle Scholar
  13. [Hu]
    L.K. Hua, Harmonic Analysis of Functions of Several Com-plex Variables in the Classical Domains, Amer. Math. Soc., Providence, RI, 1963.Google Scholar
  14. [J]
    K.D. Johnson, On a ring of invariant polynomials on a Her-mitian symmetric space, J. Algebra, 67(1980), 72–81.MathSciNetMATHCrossRefGoogle Scholar
  15. [Ka]
    K.W.J. Kadell, The Selberg-Jack symmetric functions, Preprint, 1992.Google Scholar
  16. [K1]
    A. Korányi, The Poisson integral for generalized halfplanes and bounded symmetric domains, Ann. of Math. 82(1965), 332–350.MathSciNetMATHCrossRefGoogle Scholar
  17. [K2]
    A. Korányi, The volume of symmetric domains, the Koecher Gamma function, and an integral of Selberg, Studia Sci. Math. Hungarica, 17(1982), 129–133.MATHGoogle Scholar
  18. [K3]
    A. Korányi, Hua-type integrals, hypergeometric functions and symmetric polynomials, in International Symposium in Honour of Hua Loo Keng, Vol. 2 : Analysis, Springer-Verlag and Sci. Press Beijing, 1991.Google Scholar
  19. [K4]
    A. Korányi, Holomorphic and harmonic functions on bounded symmetric domains. In: C.I.M.E. Summer Course on Geometry of Bounded Symmetric Domains, pp.126–197, Cremonese, Roma, 1968.Google Scholar
  20. [KV]
    A. Korányi and S. Vagi, Rational inner functions on bounded symmetric domains, Trans. Amer. Math. Soc. 254(1979), 179–193.MathSciNetMATHGoogle Scholar
  21. [KW]
    A. Korányi and J. A. Wolf, Realization of Hermitian symmetric spaces as generalized halfplanes, Ann. of Math. 81(1965), 265–288.MathSciNetMATHCrossRefGoogle Scholar
  22. [KS]
    B. Kostant and S. Sahi, The Capelli identity, tube do-mains, and the generalized Laplace transform, Adv. Math. 87(1991), 71–92.MathSciNetMATHGoogle Scholar
  23. [L]
    M. Lassalle, Noyau de Szegö, K-types, et algèbres de Jor-dan, C.R. Acad. Sci. Paris. 303 Série I (1986), 1–4.MATHGoogle Scholar
  24. [M]
    I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, Oxford, 1995.MATHGoogle Scholar
  25. [MT]
    N. Mok and I.-H. Tsai, Rigidity of convex realizations of irreducible bounded symmetric domains of rank ≥ 2, J. reine angew. Math. 431(1992), 91–122.MathSciNetMATHGoogle Scholar
  26. [OZ]
    B. Q. rsted and G. Zhang, Reproducing kernels and compo-sition series of vector-valued holomorphic functions on tube domains, J. Funct. Anal. 124(1994), 181–204.MathSciNetCrossRefGoogle Scholar
  27. [R]
    W. Rudin, Function Theory in the Unit Ball of Cn, Springer-Verlag, New York, 1980.CrossRefGoogle Scholar
  28. [RS]
    H. Rubenthaler and G. Schiffmann, Opérateurs différentiels de Shimura et espaces préhomogènes, Invent. Math. 90(1987), 409–442.MathSciNetMATHCrossRefGoogle Scholar
  29. [Sa]
    S. Sahi, The Capelli identity and unitary representations, Compositio Math. 81(1992), 247–260.MathSciNetMATHGoogle Scholar
  30. [S]
    I. Satake, Algebraic Structures of Symmetric Domains, Iwanami Shoten, Tokyo, and Princeton Univ. Press, Princeton, NJ, 1980.MATHGoogle Scholar
  31. [Sch]
    W. Schmid, Die Randwerte holomorpher Funktionen auf Hermitischen Räumen, Invent. Math. 9(1969), 61–80.MathSciNetMATHCrossRefGoogle Scholar
  32. [Se]
    A. Selberg, Bemerkninger om et multipelt integral, Norsk Mat. Tidsskr. 26(1944), 71–78.MathSciNetMATHGoogle Scholar
  33. [Sh]
    G. Shimura, Invariant differential operators on Hermitian symmetric spaces, Ann. Math. 132(1990), 237–272.MathSciNetMATHCrossRefGoogle Scholar
  34. [U1]
    H. Upmeier, Jordan algebras and harmonic analysis on sym-metric spaces, Amer. J. Math. 108(1986), 1–25.MathSciNetMATHCrossRefGoogle Scholar
  35. [U2]
    H. Upmeier, Toeplitz operators on bounded symmetric do-mains, Trans. Amer. Math. Soc. 280(1983), 221–237.MathSciNetMATHCrossRefGoogle Scholar
  36. [U3]
    H. Upmeier, Toeplitz C*-algebras on bounded symmetric domains, Ann. Math. 119(1984), 549–576.MathSciNetMATHCrossRefGoogle Scholar
  37. [Wa]
    N. Wallach, The analytic continuation of the discrete series, I, II, Trans. Amer. Math. Soc. 251(1979), 1–17 and 19–37.MathSciNetMATHGoogle Scholar
  38. [Wl]
    J.A. Wolf, Spaces of Constant Curvature, 5th ed., Publish or Perish, 1984.Google Scholar
  39. [W2]
    J.A. Wolf, On the classification of Hermitian symmetric spaces, J. Math. Mech. 13(1964), 489–496.MathSciNetMATHGoogle Scholar
  40. [Yl]
    Z. Yan, Duality and differential operators on the Bergman spaces of bounded symmetric domains, J. Funct. Anal. 105(1992), 171–186.MathSciNetMATHCrossRefGoogle Scholar
  41. [Y2]
    Z. Yan, Invariant differential operators and holomorphic function spaces, preprint, 1992.Google Scholar
  42. [Y3]
    Z. Yan, Differential operators and function spaces, Several Complex Variables in China, 121–142, Contemp. Math. 142, Amer. Math. Soc., Providence, RI, 1993.Google Scholar
  43. [Z1]
    K. Zhu, Duality and Hankel operators on the Bergman spaces of bounded symmetric domains, J. Funct. Anal. 81(1988), 260–278.MathSciNetMATHCrossRefGoogle Scholar
  44. [Z2]
    K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.MATHGoogle Scholar
  45. [Z3]
    K. Zhu, Harmonic analysis on bounded symmetric domains, in: M. Cheng et al. (eds.), Harmonic Analysis in China, pp.287–307, Kluwer Academic Publishers, Dordrecht,1995.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Adam Korányi
    • 1
  1. 1.Dept. Mathematics & Computer ScienceH.H. Lehman CollegeBronxUSA

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