Toeplitz operators on the Segal-Bargmann space

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

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

Toeplitz operators on the Segal-Bargmann space
HTML articles powered by AMS MathViewer

by C. A. Berger and L. A. Coburn

Trans. Amer. Math. Soc. 301 (1987), 813-829

DOI: https://doi.org/10.1090/S0002-9947-1987-0882716-4

PDF | Request permission

Abstract:

In this paper, we give a complete characterization of those functions on $2n$-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” .

References

V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 157250, DOI 10.1002/cpa.3160140303
V. Bargmann, Remarks on a Hilbert space of analytic functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 199–204. MR 133009, DOI 10.1073/pnas.48.2.199
F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134–1167 (Russian). MR 0350504

Quantization

8

Quantization in complex symmetric spaces

9

C. A. Berger and L. A. Coburn, Toeplitz operators and quantum mechanics, J. Funct. Anal. 68 (1986), no. 3, 273–299. MR 859136, DOI 10.1016/0022-1236(86)90099-6

Operator algebras and quantum statistical mechanics

Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
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 267425
Victor Guillemin, Toeplitz operators in $n$ dimensions, Integral Equations Operator Theory 7 (1984), no. 2, 145–205. MR 750217, DOI 10.1007/BF01200373
Roger Howe, Quantum mechanics and partial differential equations, J. Functional Analysis 38 (1980), no. 2, 188–254. MR 587908, DOI 10.1016/0022-1236(80)90064-6
S. C. Power, Commutator ideals and pseudodifferential $C^{\ast }$-algebras, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 124, 467–489. MR 596980, DOI 10.1093/qmath/31.4.467
Marc A. Rieffel, $C^{\ast }$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572
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-London, 1978, pp. 321–343. MR 538026
Ke He Zhu, VMO, ESV, and Toeplitz operators on the Bergman space, Trans. Amer. Math. Soc. 302 (1987), no. 2, 617–646. MR 891638, DOI 10.1090/S0002-9947-1987-0891638-4