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The Berezin Transform and Its Applications

Abstract

We give a survey on the Berezin transform and its applications in operator theory. The focus is on the Bergman space of the unit disk and the Fock space of the complex plane. The Berezin transform is most effective and most successful in the study of Hankel and Toepltiz operators.

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References

  1. Ahern P. On the range of the Berezin transform. J Funct Anal, 2004, 215: 206–216

    MathSciNet  MATH  Google Scholar 

  2. Ahern P, Flores M, Rudin W. An invariant volume-mean-value property. J Funct Anal, 1993, 111: 380–397

    MathSciNet  MATH  Google Scholar 

  3. Aleman A, Pott S, Reguera C. Sarason conjecture on the Bergman space. Int Math Res Not IMRN, 2017, 14: 4320–4349

    MathSciNet  MATH  Google Scholar 

  4. Arazy J, Englis M. Iterates and boundary behavior of the Berezin transform. Ann Inst Fourier, 2001, 51: 1101–1133

    MathSciNet  MATH  Google Scholar 

  5. Arazy J, Fisher S, Peetre J. Hankel operators on weighted Bergman spaces. Amer Math J, 1988, 110: 989–1053

    MathSciNet  MATH  Google Scholar 

  6. Arazy J, Zhang G. Invariant mean value and harmonicity in Cartan and Siegel domains//Interactions Between Functional Analysis, Harmonic Analysis, and Probability, (Columbia, MO, 1994). Lecture Notes in Pure and Applied Math 175. New York: Marcel Dekker, 1996: 19–40

    Google Scholar 

  7. Axler S, Cuckovic Z. Commuting Toeplitz operators with harmonic symbols. Integral Equations Operator Theory, 1991, 14: 1–12

    MathSciNet  MATH  Google Scholar 

  8. Axler S, Zheng D. Compact operators via the Berezin transform. Indiana Univ Math J, 1998, 47: 387–400

    MathSciNet  MATH  Google Scholar 

  9. Axler S, Zheng D. The Berezin transform on the Toeplitz algebra. Studia Math, 1998, 127: 113–136

    MathSciNet  MATH  Google Scholar 

  10. Bauer W, Coburn L, Isralowitz J. Heat flow, BMO, and compactness of Toeplitz operators. J Funct Anal, 2010, 259: 57–78

    MathSciNet  MATH  Google Scholar 

  11. Bauer W, Isralowitz J. Compactness characterization of operators in the Toeplitz algebra of the Fock space F pα . J Funct Anal, 2012, 263: 1323–1355

    MathSciNet  MATH  Google Scholar 

  12. Bekolle D, Berger C, Coburn L, Zhu K. BMO in the Bergman metric on bounded symmetric domains. J Funct Anal, 1990, 93: 64–89

    MathSciNet  MATH  Google Scholar 

  13. Berezin F. Wick and anti-Wick symbols of operators (Russian). Mat Sb, 1971, 86: 578–610

    MathSciNet  Google Scholar 

  14. Berezin F. Covariant and contra-variant symbols of operators. Math USSR-Izv, 1972, 6: 1117–1151

    Google Scholar 

  15. Berezin F. Quantization. Math USSR-Izv, 1974, 8: 1109–1163

    MATH  Google Scholar 

  16. Berezin F. Quantization of complex symmetric spaces (Russian). Izv Akad Ser Mat, 1975, 39: 363–402

    MATH  Google Scholar 

  17. Berezin F. General concept of quantization. Comm Math Phys, 1975, 40: 153–174

    MathSciNet  MATH  Google Scholar 

  18. Berezin F. Introduction to Superanalysis. Dordrecht: Reidel, 1987

    MATH  Google Scholar 

  19. Berger C, Coburn L. Toeplitz operators and quantum mechanics. J Funct Anal, 1986, 68: 273–299

    MathSciNet  MATH  Google Scholar 

  20. Berger C, Coburn L. Toeplitz operators on the Segal-Bargmann space. Trans Amer Math Soc, 1987, 301: 813–829

    MathSciNet  MATH  Google Scholar 

  21. Berger C, Coburn L. Heat flow and Berezin-Toeplitz estimates. Amer J Math, 1994, 116: 563–590

    MathSciNet  MATH  Google Scholar 

  22. Berger C, Coburn L, Zhu K. BMO in the Bergman metric on the classical domains. Bull Amer Math Soc, 1987, 17: 133–136

    MathSciNet  MATH  Google Scholar 

  23. Berger C, Coburn L, Zhu K. Function theory on Cartan domains and Berezin-Toeplitz symbol calculus. Amer J Math, 1988, 110: 921–953

    MathSciNet  MATH  Google Scholar 

  24. Bommier-Hato H. Lipschitz estimates for the Berezin transform. J Funct Spaces Appl, 2010, 8: 103–128

    MathSciNet  MATH  Google Scholar 

  25. Bommier-Hato H. Derivatives of the Berezin transform. J Funct Spaces Appl, 2012, 15 pages

  26. Bommier-Hato H, Youssfi E, Zhu K. Sarason’s Toeplitz product problem for a class of Fock spaces. Bull Sci Math, 2017, 141: 408–442

    MathSciNet  MATH  Google Scholar 

  27. H. Cho, J. Park, and K. Zhu, Products of Toeplitz operators on the Fock space. Proc Amer Math Soc, 2014, 142: 2483–2489

    MathSciNet  MATH  Google Scholar 

  28. Coburn L. A Lipschitz estimate for Berezin’s operator calculus. Proc Amer Math Soc, 2005, 133: 127–131

    MathSciNet  MATH  Google Scholar 

  29. Coburn L. Sharp Berezin-Lipschitz estimates. Proc Amer Math Soc, 2007, 135: 1163–1168

    MathSciNet  MATH  Google Scholar 

  30. Coburn L. Berezin-Toeplitz quantization//Algebraic Methods in Operator Theory. Boston: Birkhauser, 1994: 101–108

    Google Scholar 

  31. Coburn L, Isralowitz J, Li B. Toeplitz operators with BMO symbols on the Segal-Bargmann space. Trans Amer Math Soc, 2011, 363: 3015–3030

    MathSciNet  MATH  Google Scholar 

  32. Coburn L, Li B. Directional derivative estimates for Berezin’s operator calculus. Proc Amer Math Soc, 2008, 136: 641–649

    MathSciNet  MATH  Google Scholar 

  33. Cuckovic Z, Li B. Berezin Transform, Mellin Transform and Toeplitz Operators. Complex Anal Oper Theory, 2012, 6: 189–218

    MathSciNet  MATH  Google Scholar 

  34. Davidson K, Douglas R. The generalized Berezin transform and commutator ideals. Pacific J Math, 2005, 222: 29–56

    MathSciNet  MATH  Google Scholar 

  35. Duren P, Schuster A. Bergman Spaces. American Mathematical Society, 2004

  36. Englis M. Functions invariant under the Berezin transform. J Funct Anal, 1994, 121: 233–254

    MathSciNet  MATH  Google Scholar 

  37. Englis M. Toeplitz operators and the Berezin transform on H2. Linear Alg Appl, 1995, 223/224: 171–204

    MATH  Google Scholar 

  38. Englis M. Berezin transform and the Laplace-Beltrami operator. Algebra i Analiz, 1995, 7: 176–195

    MathSciNet  MATH  Google Scholar 

  39. Englis M. Asymptotics of the Berezin transform and quantization on planar domains. Duke Math J, 1995, 79: 57–76

    MathSciNet  MATH  Google Scholar 

  40. Englis M. Berezin quantization and reproducing kernels on complex domains. Trans Amer Math Soc, 1996, 348: 411–479

    MathSciNet  MATH  Google Scholar 

  41. Englis M. Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integral Equations Operator Theory, 1999, 33: 426–455

    MathSciNet  MATH  Google Scholar 

  42. Englis M, Otáhalová R. Covariant derivatives of the Berezin transform. Trans Amer Math Soc, 2011, 363: 5111–5129

    MathSciNet  MATH  Google Scholar 

  43. Englis M, Zhang G. On the derivatives of the Berezin transform. Proc Amer Math Soc, 2006, 134: 2285–2294

    MathSciNet  MATH  Google Scholar 

  44. Le Floch Y. A Brief Introduction to Berezin-Toeplitz Operators on Compact Kähler Manifolds. CRM Short Courses. Springer, 2018

  45. Garnett J. Bounded Analytic Functions. New York: Academic Press, 1981

    MATH  Google Scholar 

  46. Hedenmalm H, Korenblum B, Zhu K. Theory of Bergman Spaces. New York: Springer-Verlag, 2000

    MATH  Google Scholar 

  47. Ioos L, Kaminker V, Polterovich L, Shmoish D. Spectral aspects of the Berezin transform. preprint, 2020

  48. Ioos L, Lu W, Ma X, Marinescu G. Berezin-Toeplitz quantization for eigenstates of the Bochner-Laplacian on symplectic manifolds. J Geom Anal, 2020, 30: 2615–2646

    MathSciNet  MATH  Google Scholar 

  49. Janson S, Peetre J, Rochberg R. Hankel forms and the Fock space. Revista Mat Ibero-Amer, 1987, 3: 58–80

    MathSciNet  MATH  Google Scholar 

  50. Karabegov A, Schlichenmaier M. Identification of Berezin-Toeplitz deformation quantization. J Reine Angew Math, 2001, 540: 49–76

    MathSciNet  MATH  Google Scholar 

  51. Kilic S. The Berezin symbol and multipliers of functional Hilbert spaces. Proc Amer Math Soc, 1995, 123: 3687–3691

    MathSciNet  MATH  Google Scholar 

  52. Korenblum B, Zhu K H. An application of Tauberian theorems to Toeplitz operators. J Operator Theory, 1995, 33: 353–361

    MathSciNet  MATH  Google Scholar 

  53. Lee J. Properties of the Berezin transform of bounded functions. Bull Austral Math Soc, 1999, 59: 21–31

    MathSciNet  MATH  Google Scholar 

  54. Li B. The Berezin transform and Laplace-Beltrami operator. J Math Anal Appl, 2007, 327: 1155–1166

    MathSciNet  MATH  Google Scholar 

  55. Li B. The Berezin transform and m-th order Bergman metric. Trans Amer Math Soc, 2011, 363: 3031–3056

    MathSciNet  MATH  Google Scholar 

  56. Luecking D, Zhu K. Composition operators belonging to the Schatten ideals. Amer J Math, 1992, 114: 1127–1145

    MathSciNet  MATH  Google Scholar 

  57. Ma X, Marinescu G. Berezin-Toeplitz quantization on Kähler manifolds. J Reine Angew Math, 2012, 662: 1–56

    MathSciNet  MATH  Google Scholar 

  58. Ma P, Yan F, Zheng D, Zhu K. Products of Hankel operators on the Fock space. J Funct Anal, 2019, 277: 2644–2663

    MathSciNet  MATH  Google Scholar 

  59. Ma P, Yan F, Zheng D, Zhu K. Mixed products of Hankel and Toeplitz operators on the Fock space. J Operator Theory, 2020, 84: 35–47

    MathSciNet  MATH  Google Scholar 

  60. MacCluer B. Compact composition operators on Hp(Bn). Mich Math J, 1985, 32: 237–248

    MATH  Google Scholar 

  61. MacCluer B, Shapiro J. Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Canadian Math J, 1986, 38: 878–906

    MathSciNet  MATH  Google Scholar 

  62. Nam K, Zheng D, Zhong C. m-Berezin transform and compact operators. Rev Mat Iberoam, 2006, 22: 867–892

    MathSciNet  MATH  Google Scholar 

  63. Nazarov F. A counterexample to Sarason’s conjecture. preprint, 1997

  64. Nordgren E, Rosenthal P. Boundary values of Berezin symbols. Operator Theory Advances and Applications, 1994, 73: 362–368

    MathSciNet  MATH  Google Scholar 

  65. Peetre J. The Berezin transform and Ha-plitz operators. J Operator Theory, 1990, 24: 165–186

    MathSciNet  MATH  Google Scholar 

  66. Rao N V. The range of the Berezin transform. J Math Sci (NY), 2018, 228(6): 684–694

    MathSciNet  MATH  Google Scholar 

  67. Sarason D. Products of Toeplitz operators//Havin V P, Nikolski N K, eds. Linear and Complex Analysis Problem Book 3, Part I, Lecture Notes in Math 1573. Berlin: Springer, 1994: 318–319

    Google Scholar 

  68. Schlichenmaier M. Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results. Adv Math Phys, 2010: 927280

  69. Shapiro J. The essential norm of a composition operator. Ann of Math, 1987, 12: 375–404

    MathSciNet  MATH  Google Scholar 

  70. Shklyarov D, Zhang G. The Berezin transform on the quantum unit ball. J Math Physics, 2003, 44(4344): 4344–4373

    MathSciNet  MATH  Google Scholar 

  71. Stroethoff K. The Berezin transform and operators on spaces of analytic functions//Linear operators (Warsaw, 1994), Banach Center Publ 38. Warsaw: Polish Acad Sci, 1997: 361–380

    MATH  Google Scholar 

  72. Stroethoff K, Zheng D. Toeplitz and Hankel operators on Bergman spaces. Trans Amer Math Soc, 1992, 329: 773–794

    MathSciNet  MATH  Google Scholar 

  73. Stroethoff K, Zheng D. Products of Hankel and Toeplitz operators on the Bergman space. J Funct Anal, 1999, 169: 289–313

    MathSciNet  MATH  Google Scholar 

  74. Suarez D. Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev Mat Iberoam, 2004, 20: 563–610

    MathSciNet  MATH  Google Scholar 

  75. Suarez D. Approximation and the n-Berezin transform of operators on the Bergman space. J Reine Angew Math, 2005, 581: 175–192

    MathSciNet  MATH  Google Scholar 

  76. Suarez D. The essential norm of operators in the Toeplitz algebra on Ap(Bn). Indiana Univ Math J, 2007, 56: 2185–2232

    MathSciNet  MATH  Google Scholar 

  77. Untenberger A, Upmeier H. The Berezin transform and invariant differential operators. Comm Math Phys, 1994, 164: 563–597

    MathSciNet  MATH  Google Scholar 

  78. Zhang G. Berezin transform on compact Hermitian symmetric spaces. Manuscripta Math, 1998, 97: 371–388

    MathSciNet  MATH  Google Scholar 

  79. Zheng D. Hankel operators and Toeplitz operators on the Bergman space. J Funct Anal, 1989, 83: 98–120

    MathSciNet  MATH  Google Scholar 

  80. Zheng D. The distribution function inequality and products of Toeplitz operators and Hankel operators. J Funct Anal, 1996, 138: 477–501

    MathSciNet  MATH  Google Scholar 

  81. Zhu K. VMO, ESV, and Toeplitz operators on the Bergman space. Trans Amer Math Soc, 1987, 302: 617–646

    MathSciNet  MATH  Google Scholar 

  82. Zhu K. Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J Operator Theory, 1988, 20: 329–357

    MathSciNet  MATH  Google Scholar 

  83. Zhu K. Schatten class Hankel operators on the Bergman space of the unit ball. Amer J Math, 1991, 113: 147–167

    MathSciNet  MATH  Google Scholar 

  84. Zhu K. Operator Theory in Function Spaces. American Mathematical Society, 2007

  85. Zhu K. Analysis on Fock Spaces. New York: Springer, 2012

    MATH  Google Scholar 

  86. Zorboska N. The Berezin transform and radial operators. Proc Amer Math Soc, 2002, 131: 793–800

    MathSciNet  MATH  Google Scholar 

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Correspondence to Kehe Zhu.

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Dedicated to the memory of Professor Jiarong YU

Research partially supported by NNSF of China (11720101003), NSF of Guangdong Province (2018A030313512), Key projects of fundamental research in universities of Guangdong Province (2018KZDXM034) and STU Scientific Research Foundation (NTF17009).

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Zhu, K. The Berezin Transform and Its Applications. Acta Math Sci 41, 1839–1858 (2021). https://doi.org/10.1007/s10473-021-0603-5

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  • DOI: https://doi.org/10.1007/s10473-021-0603-5

Key words

  • reproducing kernel
  • reproducing kernel Hilbert space
  • Hardy space
  • Bergman space
  • Fock space
  • Berezin transform
  • Toeplitz operator
  • Hankel operator
  • composition operator

2010 MR Subject Classification

  • 30H10
  • 30H20
  • 32A35
  • 32A36
  • 46E22
  • 47B32
  • 47B33
  • 47B35
  • 47G10