Asymptotic expansions for Toeplitz operators on symmetric spaces of general type

Engliš, Miroslav; Upmeier, Harald

doi:10.1090/S0002-9947-2014-06130-8

Asymptotic expansions for Toeplitz operators on symmetric spaces of general type
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by Miroslav Engliš and Harald Upmeier PDF

Trans. Amer. Math. Soc. 367 (2015), 423-476 Request permission

Abstract:

A general theory of Berezin-Toeplitz quantization for symmetric spaces is presented, with emphasis on representation-theoretic asymptotic expansions, which applies to spaces of compact and non-compact type, both in the classical setting of hermitian symmetric spaces and also for their real forms. The Berezin (or Wick type) calculus and its opposite “anti-Wick” type calculus are treated on an equal footing.

References

S. Twareque Ali and Miroslav Engliš, Quantization methods: a guide for physicists and analysts, Rev. Math. Phys. 17 (2005), no. 4, 391–490. MR 2151954, DOI 10.1142/S0129055X05002376
Jonathan Arazy and Bent Ørsted, Asymptotic expansions of Berezin transforms, Indiana Univ. Math. J. 49 (2000), no. 1, 7–30. MR 1777039, DOI 10.1512/iumj.2000.49.1815
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
David Borthwick, Andrzej Lesniewski, and Harald Upmeier, Nonperturbative deformation quantization of Cartan domains, J. Funct. Anal. 113 (1993), no. 1, 153–176. MR 1214901, DOI 10.1006/jfan.1993.1050
Martin Bordemann, Eckhard Meinrenken, and Martin Schlichenmaier, Toeplitz quantization of Kähler manifolds and $\textrm {gl}(N)$, $N\to \infty$ limits, Comm. Math. Phys. 165 (1994), no. 2, 281–296. MR 1301849, DOI 10.1007/BF02099772
Benjamin Cahen, Berezin quantization for holomorphic discrete series representations: the non-scalar case, Beitr. Algebra Geom. 53 (2012), no. 2, 461–471. MR 2971754, DOI 10.1007/s13366-011-0066-2
Miroslav Engliš, Weighted Bergman kernels and quantization, Comm. Math. Phys. 227 (2002), no. 2, 211–241. MR 1903645, DOI 10.1007/s002200200634
Miroslav Engliš, Berezin-Toeplitz quantization on the Schwartz space of bounded symmetric domains, J. Lie Theory 15 (2005), no. 1, 27–50. MR 2115226
Miroslav Engliš and Harald Upmeier, Toeplitz quantization and asymptotic expansions for real bounded symmetric domains, Math. Z. 268 (2011), no. 3-4, 931–967. MR 2818737, DOI 10.1007/s00209-010-0702-9
Miroslav Engliš and Harald Upmeier, Toeplitz quantization and asymptotic expansions: geometric construction, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 021, 30. MR 2481471, DOI 10.3842/SIGMA.2009.021
Miroslav Engliš and Harald Upmeier, Toeplitz quantization and asymptotic expansions: Peter-Weyl decomposition, Integral Equations Operator Theory 68 (2010), no. 3, 427–449. MR 2735445, DOI 10.1007/s00020-010-1808-5
Miroslav Engliš and Harald Upmeier, Real Berezin transform and asymptotic expansion for symmetric spaces of compact and non-compact type, Recent progress in operator theory and its applications, Oper. Theory Adv. Appl., vol. 220, Birkhäuser/Springer Basel AG, Basel, 2012, pp. 97–114. MR 2953873, DOI 10.1007/978-3-0348-0346-5_{6}
Miroslav Engliš and Genkai Zhang, Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces, Math. Z. 264 (2010), no. 4, 901–912. MR 2593299, DOI 10.1007/s00209-009-0495-x
J. Faraut and A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), no. 1, 64–89. MR 1033914, DOI 10.1016/0022-1236(90)90119-6
Frank Forelli, Measures whose Poisson integrals are pluriharmonic, Illinois J. Math. 18 (1974), 373–388. MR 342723
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Alexander V. Karabegov and Martin Schlichenmaier, Identification of Berezin-Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49–76. MR 1868597, DOI 10.1515/crll.2001.086
Ewa Ligocka, On the Forelli-Rudin construction and weighted Bergman projections, Studia Math. 94 (1989), no. 3, 257–272. MR 1019793, DOI 10.4064/sm-94-3-257-272
O. Loos, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977.
Martin Schlichenmaier, Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization, Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, pp. 289–306. MR 1805922
A. Unterberger and H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), no. 3, 563–597. MR 1291245, DOI 10.1007/BF02101491
Genkai Zhang, Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3769–3787. MR 1837258, DOI 10.1090/S0002-9947-01-02832-X
Genkai Zhang, Berezin transform on compact Hermitian symmetric spaces, Manuscripta Math. 97 (1998), no. 3, 371–388. MR 1654800, DOI 10.1007/s002290050109
Genkai Zhang, Branching coefficients of holomorphic representations and Segal-Bargmann transform, J. Funct. Anal. 195 (2002), no. 2, 306–349. MR 1940358, DOI 10.1006/jfan.2002.3957
Genkai Zhang, Degenerate principal series representations and their holomorphic extensions, Adv. Math. 223 (2010), no. 5, 1495–1520. MR 2592500, DOI 10.1016/j.aim.2009.09.014