Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 101, 6 pages      arXiv:0710.4995
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Toeplitz Operators, Kähler Manifolds, and Line Bundles

Tatyana Foth
Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada

Received August 23, 2007, in final form October 23, 2007; Published online October 26, 2007

This is a survey paper. We discuss Toeplitz operators in Kähler geometry, with applications to geometric quantization, and review some recent developments.

Key words: Kähler manifolds; holomorphic line bundles; geometric quantization; Toeplitz operators.

pdf (209 kb)   ps (154 kb)   tex (10 kb)


  1. Berezin F., Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134-1167 (in Russian).
  2. Berezin F., Spectral properties of generalized Toeplitz matrices, Mat. Sb. (N.S.) 95 (137) (1974), 305-325, 328 (in Russian).
  3. Berezin F., General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174.
  4. Berman R., Berndtsson B., Sjoestrand J., Asymptotics of Bergman kernels, math.CV/0506367.
  5. Bloch A., Golse F., Uribe A., Dispersionless Toda and Toeplitz operators, Duke Math. J. 117 (2003), 157-196.
  6. Bordemann M., Meinrenken E., Schlichenmaier M., Toeplitz quantization of Kähler manifolds and gl(N), N limits, Comm. Math. Phys. 165 (1994), 281-296.
  7. Borthwick D., Introduction to Kähler quantization, in Quantization, the Segal-Bargmann transform and semiclassical analysis, 1st Summer School in Analysis and Mathematical Physics (Mexico, 1998), Contemp. Math. 260 (2000), 91-132.
  8. Borthwick D., Lesniewski A., Upmeier H., Nonperturbative deformation quantization of Cartan domains, J. Funct. Anal. 113 (1993), 153-176.
  9. Borthwick D., Paul T., Uribe A., Semiclassical spectral estimates for Toeplitz operators, Ann. Inst. Fourier (Grenoble) 48 (1998), 1189-1229.
  10. Borthwick D., Paul T., Uribe A., Legendrian distributions with applications to relative Poincaré series, Invent. Math. 122 (1995), 359-402, hep-th/9406036.
  11. Böttcher A., Silbermann B., Analysis of Toeplitz operators, Springer-Verlag, Berlin, 1990.
  12. Boutet de Monvel L., On the index of Toeplitz operators of several complex variables, Invent. Math. 50 (1978/79), 249-272.
  13. Boutet de Monvel L., Toeplitz operators - an asymptotic quantization of symplectic cones, in Stochastic Processes and Their Applications in Mathematics and Physics (Bielefeld, 1985), Math. Appl., Vol. 61, Kluwer Acad. Publ., Dordrecht, 1990, 95-106.
  14. Boutet de Monvel L., Logarithmic trace of Toeplitz projectors, Math. Res. Lett. 12 (2005), 401-412, math.CV/0412252.
  15. Boutet de Monvel L., Guillemin V., The spectral theory of Toeplitz operators, Annals of Math. Studies, Vol. 99, Princeton University Press, Princeton, New Jersey, 1981.
  16. Boutet de Monvel L., Sjöstrand J., Sur la singularite des noyaux de Bergman et de Szego, in Journées: Équations aux Dérivees Partielles de Rennes (1975), Asterisque, no. 34-35 (1976), 123-164.
  17. Brown A., Halmos P., Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89-102.
  18. Charles L., Berezin-Toeplitz operators, a semi-classical approach, Comm. Math. Phys. 239 (2003), 1-28.
  19. Charles L., Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Comm. Partial Differential Equations 28 (2003), 1527-1566.
  20. Charles L., Toeplitz operators and Hamiltonian torus actions, J. Funct. Anal. 236 (2006), 299-350, math.SG/0405128.
  21. Charles L., Symbolic calculus for Toeplitz operators with half-form, J. Symplectic Geom. 4 (2006), 171-198, math.SG/0602167.
  22. Charles L., Semi-classical properties of geometric quantization with metaplectic correction, Comm. Math. Phys. 270 (2007), 445-480, math.SG/0602168.
  23. Donaldson S., Remarks on gauge theory, complex geometry and 4-manifold topology, in Fields Medallists' Lectures, World Sci. Ser. 20th Century Math., Vol. 5, World Sci. Publ., River Edge, NJ, 1997, 384-403.
  24. Douglas R., Ideals in Toeplitz algebras, Houston J. Math. 31 (2005), 529-539, math.OA/0309438.
  25. Douglas R., Widom H., Toeplitz operators with locally sectorial symbols, Indiana Univ. Math. J. 20 (1970/1971), 385-388.
  26. Foth T., Toeplitz operators, deformations, and asymptotics, J. Geom. Phys. 57 (2007), 855-861.
  27. Foth T., Uribe A., The manifold of compatible almost complex structures and geometric quantization, Comm. Math. Phys., to appear.
  28. Fujiki A., Moduli space of polarized algebraic manifolds and Kähler metrics, Sugaku Expositions 5 (1992), 173-191.
  29. Guillemin V., Star products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys. 35 (1995), 85-89.
  30. Jewell N., Toeplitz operators on the Bergman spaces and in several complex variables, Proc. London Math. Soc. (3) 41 (1980), 193-216.
  31. Jewell N., Krantz S., Toeplitz operators and related function algebras on certain pseudoconvex domains, Trans. Amer. Math. Soc. 252 (1979), 297-312.
  32. Karabegov A., Schlichenmaier M., Identification of Berezin-Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49-76, math.QA/0006063.
  33. Klein E., The numerical range of a Toeplitz operator, Proc. Amer. Math. Soc. 35 (1972), 101-103.
  34. Klein E., More algebraic properties of Toeplitz operators, Math. Ann. 202 (1973), 203-207.
  35. Klimek S., Lesniewski A., Quantum Riemann surfaces. I. The unit disc, Comm. Math. Phys. 146 (1992), 103-122.
    Klimek S., Lesniewski A., Quantum Riemann surfaces. II. The discrete series, Lett. Math. Phys. 24 (1992), 125-139.
    Klimek S., Lesniewski A., Quantum Riemann surfaces. III. The exceptional cases, Lett. Math. Phys. 32 (1994), 45-61.
  36. Klimek S., Lesniewski A., Quantum Riemann surfaces for arbitrary Planck's constant, J. Math. Phys. 37 (1996), 2157-2165.
  37. Kostant B., Quantization and unitary representations. I. Prequantization, in Lectures in Modern Analysis and Applications. III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, 87-208.
  38. Peller V., Invariant subspaces of Toeplitz operators with piecewise continuous symbols, Proc. Amer. Math. Soc. 119 (1993), 171-178.
  39. Reshetikhin N., Takhtajan L., Deformation quantization of Kähler manifolds, in L.D. Faddeev's Seminar on Mathematical Physics,, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Amer. Math. Soc., Providence, RI, 2000, 257-276, math.QA/9907171.
  40. Salinas N., Sheu A., Upmeier H., Toeplitz operators on pseudoconvex domains and foliation C*-algebras, Ann. Math. (2) 130 (1989), 531-565.
  41. Schlichenmaier M., Berezin-Toeplitz quantization of compact Kähler manifolds, in Quantization, Coherent States, and Poisson Structures (Bialowieza, 1995), PWN, Warsaw, 1998, 101-115.
  42. Schlichenmaier M., Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization, in Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., Vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, 289-306, math.QA/9910137.
  43. Schlichenmaier M., Berezin-Toeplitz quantization and Berezin transform, in Long Time Behaviour of Classical and Quantum Systems (Bologna, 1999), Ser. Concr. Appl. Math., Vol. 1, World Sci. Publ., River Edge, NJ, 2001, 271-287.
  44. Souriau J.-M., Structure of dynamical systems, A symplectic view of physics, Progress in Mathematics, Vol. 149, Birkhäuser Boston, Inc., Boston, MA, 1997 (translation of Structure des systèmes dynamiques, Maîtrises de mathématiques Dunod, Paris, 1970).
  45. Stroethoff K., Zheng D., Bounded Toeplitz products on the Bergman space of the polydisk, J. Math. Anal. Appl. 278 (2003), 125-135.
  46. Stroethoff K., Zheng D., Bounded Toeplitz products on Bergman spaces of the unit ball, J. Math. Anal. Appl. 325 (2007), 114-129.
  47. Upmeier H., Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280 (1983), 221-237.
  48. Upmeier H., Toeplitz C*-algebras on bounded symmetric domains, Ann. Math. (2) 119 (1984), 549-576.
  49. Upmeier H., Toeplitz operators on symmetric Siegel domains, Math. Ann. 271 (1985), 401-414.
  50. Upmeier H., Toeplitz operators and index theory in several complex variables, Operator Theory: Advances and Applications, Vol. 81, Birkhäuser Verlag, Basel, 1996.
  51. Widom H., On the spectrum of a Toeplitz operator, Pacific J. Math. 14 (1964), 365-375.
  52. Widom H., Toeplitz operators on Hp, Pacific J. Math. 19 (1966), 573-582.
  53. Zelditch S., Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 (1997), 305-363, math-ph/0002007.
  54. Zelditch S., Quantum maps and automorphisms, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., Vol. 232, Birkhäuser Boston, Boston, MA, 2005, 623-654, math.QA/0307175.
  55. Zelditch S., Quantum dynamics from the semi-classical point of view, unpublished notes, available at

Previous article   Next article   Contents of Volume 3 (2007)