×

zbMATH — the first resource for mathematics

The Berezin transform and invariant differential operators. (English) Zbl 0843.32019
The authors study the Berezin calculus of operators, particularly the Berezin transform, on any bounded, symmetric domain. They are able to express the Berezin transform in terms of certain invariant differential operators.
Along the way, the authors provide a nice introduction to reproducing kernels, Berezin theory, mathematical quantization, and analysis on bounded symmetric domains.
As well as providing a new point of view on the Berezin transform, this paper will serve as a useful resource.

MSC:
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
47N50 Applications of operator theory in the physical sciences
32A30 Other generalizations of function theory of one complex variable
43A85 Harmonic analysis on homogeneous spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [B1] Berezin, F.A.: Quantization. Math. USSR Izvestija8, 1109–1165 (1974) · Zbl 0312.53049
[2] [B2] Berezin, F.A.: Quantization in complex symmetric spaces. Math. USSR Izvestija9, 341–378 (1975) · Zbl 0324.53049
[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) · Zbl 0439.47038
[4] [BK] Braun, H., Koecher, M.: Jordan-Algebren. Berlin, Heidelberg, New York: Springer 1966
[5] [BLU] Borthwick, D., Lesniewski, A., Upmeier, H.: Non-perturbative deformation quantization of Cartan domains. J. Funct. Anal.113, 153–176 (1993) · Zbl 0794.46051
[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) · Zbl 0765.32005
[7] [CGR] Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds II. Trans. Am. Math. Soc. (to appear) · Zbl 0788.53062
[8] [F1] Faraut, J.: Jordan algebras and symmetric cones. Oxford University Press 1994
[9] [FK] Faraut, J., Korányi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal.88, 64–89 (1990) · Zbl 0718.32026
[10] [H1] Helgason, S.: A duality in integral geometry on symmetric spaces. Proc. US-Japan Seminar Kyoto (1965), Tokyo: Nippon Hyronsha 1966
[11] [H2] Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962 · Zbl 0111.18101
[12] [H3] Helgason, S.: Groups and geometric analysis. New York: Academic Press 1984
[13] [K1] Karpelevic, F.I.: Orispherical radial parts of Laplace operators on symmetric spaces. Sov. Math.3, 528–531 (1962)
[14] [KS1] Korányi, A., Stein, E.:H 2 spaces of generalized half-planes. Studia Math.14, 379–388 (1972) · Zbl 0224.32004
[15] [KS2] Kostant, B., Sahi, S.: The Capelli identity, tube domains, and the generalized Laplace transform. Adv. Math.87, 71–92 (1991) · Zbl 0748.22008
[16] [KU] Kaup, W., Upmeier, H.: Jordan algebras and symmetric Siegel domains in Banach spaces. Math. Z.157, 179–200 (1977) · Zbl 0357.32018
[17] [L1] Loos, O.: Bounded symmetric domains and Jordan pairs. Univ. of California, Irvine 1977
[18] [L2] Loos, O.: Jordan Paris. Lect. Notes in Math.460, Berlin: Springer-Verlag 1975
[19] [O1] Oersted, B.: A model for an interacting quantum field. J. Funct. Anal.36, 53–71 (1980) · Zbl 0504.46057
[20] [P1] Perelomov, A.: Generalized coherent states and their applications. Berlin, Heidelberg, New York: Springer 1986 · Zbl 0605.22013
[21] [P2] Peetre, J.: The Berezin transform and Haplitz operators. J. Oper. Th.24, 165–186 (1990) · Zbl 0793.47026
[22] [RV] Rossi, H., Vergne, M.: Analytic continuation of the holomorphic discrete series of a semisimple Lie group, Acta Math.136, 1–59 (1976) · Zbl 0356.32020
[23] [S1] Schmid, W.: Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen. Invent. Math.9, 61–80 (1969) · Zbl 0219.32013
[24] [U1] Upmeier, H.: ToeplitzC *-algebras on bounded symmetric domains. Ann. Math.119, 549–576 (1984) · Zbl 0549.46031
[25] [U2] Unterberger, A.: Quantification de certains espaces hermitiens symétriques. Séminaire Goulaouic-Schwartz (1979–80), Ecole Polytechnique, Paris
[26] [U3] Upmeier, H.: JordanC *-algebras and symmetric Banach manifolds. Amsterdam: North-Holland 1985
[27] [U4] Upmeier, H.: Jordan algebras in analysis, operator theory and quantum mechanics. CBMS Conference Series 67 (1987)
[28] [U5] Upmeier, H.: Jordan algebras and harmonic analysis on symmetric spaces. Am. J. Math.108, 1–25 (1986) · Zbl 0603.46055
[29] [U6] Upmeier, H.: Toeplitz operators on bounded symmetric domains. Trans. Am. Math. Soc.280, 221–237 (1983) · Zbl 0527.47019
[30] [U7] Unterberger, A., Unterberger, J.: Quantification et analyse pseudo-différentielle. Ann. Scient. Éc. Norm. Sup.21, 133–158 (1988) · Zbl 0646.58025
[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) · Zbl 0561.35082
[32] [W] Wallach, N.: The analytic continuation of the discrete series. Trans. Am. Math. Soc.251, 1–37 (1979) · Zbl 0419.22018
[33] [Y1] Yau, S.-T.: Uniformization of geometric structures. Proc. Symp. Pure Math.48, 265–274 (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.