Toeplitz operators on the Segal-Bargmann space

Berger, C. A.; Coburn, L. A.

doi:10.1090/S0002-9947-1987-0882716-4

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Toeplitz operators on the Segal-Bargmann space

Authors: C. A. Berger and L. A. Coburn
Journal: Trans. Amer. Math. Soc. 301 (1987), 813-829
MSC: Primary 47B35; Secondary 81D07
MathSciNet review: 882716
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Abstract: In this paper, we give a complete characterization of those functions on -dimensional Euclidean space for which the Berezin-Toeplitz quantizations admit a symbol calculus modulo the compact operators. The functions in question are characterized by a condition of ``small oscillation at infinity'' .

[1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 0157250 (28 #486)
[2] V. Bargmann, Remarks on a Hilbert space of analytic functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 199–204. MR 0133009 (24 #A2845)
[3] F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134–1167 (Russian). MR 0350504 (50 #2996)
[4] -, Quantization, Math. USSR Izv. 8 (1974), 1109-1163.
[5] -, Quantization in complex symmetric spaces, Math. USSR Izv. 9 (1975), 341-379.
[6] C. A. Berger and L. A. Coburn, Toeplitz operators and quantum mechanics, J. Funct. Anal. 68 (1986), no. 3, 273–299. MR 859136 (88b:46098), http://dx.doi.org/10.1016/0022-1236(86)90099-6
[7] O. Bratteli and D. Robinson, Operator algebras and quantum statistical mechanics I, II, Springer, 1979, 1981.
[8] Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964. MR 0181836 (31 #6062)
[9] A. Grossmann, G. Loupias, and E. M. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 2, 343–368, viii (1969) (English, with French summary). MR 0267425 (42 #2327)
[10] Victor Guillemin, Toeplitz operators in 𝑛 dimensions, Integral Equations Operator Theory 7 (1984), no. 2, 145–205. MR 750217 (86i:58130), http://dx.doi.org/10.1007/BF01200373
[11] Roger Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), no. 2, 188–254. MR 587908 (83b:35166), http://dx.doi.org/10.1016/0022-1236(80)90064-6
[12] S. C. Power, Commutator ideals and pseudodifferential 𝐶*-algebras, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 124, 467–489. MR 596980 (82c:47033), http://dx.doi.org/10.1093/qmath/31.4.467
[13] Marc A. Rieffel, 𝐶*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572 (83b:46087)
[14] I. E. Segal, Lectures at the Summer Seminar on Applied Math., Boulder, Col., 1960.
[15] I. E. Segal, The complex-wave representation of the free boson field, Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., vol. 3, Academic Press, New York, 1978, pp. 321–343. MR 538026 (82d:81069)
[16] Ke He Zhu, VMO, ESV, and Toeplitz operators on the Bergman space, Trans. Amer. Math. Soc. 302 (1987), no. 2, 617–646. MR 891638 (89a:47038), http://dx.doi.org/10.1090/S0002-9947-1987-0891638-4

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