THE BEREZIN TRANSFORM AND ITS APPLICATIONS

doi:10.1007/s10473-021-0603-5

Abstract

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.

Key words: reproducing kernel, reproducing kernel Hilbert space, Hardy space, Bergman space, Fock space, Berezin transform, Toeplitz operator, Hankel operator, composition operator

CLC Number:

30H10

Kehe ZHU. THE BEREZIN TRANSFORM AND ITS APPLICATIONS[J].Acta mathematica scientia,Series B, 2021, 41(6): 1839-1858.

Trendmd

References

[1] Ahern P. On the range of the Berezin transform. J Funct Anal, 2004, 215:206-216
[2] Ahern P, Flores M, Rudin W. An invariant volume-mean-value property. J Funct Anal, 1993, 111:380-397
[3] Aleman A, Pott S, Reguera C. Sarason conjecture on the Bergman space. Int Math Res Not IMRN, 2017, 14:4320-4349
[4] Arazy J, Englis M. Iterates and boundary behavior of the Berezin transform. Ann Inst Fourier, 2001, 51:1101-1133
[5] Arazy J, Fisher S, Peetre J. Hankel operators on weighted Bergman spaces. Amer Math J, 1988, 110:989-1053
[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
[7] Axler S, Cuckovic Z. Commuting Toeplitz operators with harmonic symbols. Integral Equations Operator Theory, 1991, 14:1-12
[8] Axler S, Zheng D. Compact operators via the Berezin transform. Indiana Univ Math J, 1998, 47:387-400
[9] Axler S, Zheng D. The Berezin transform on the Toeplitz algebra. Studia Math, 1998, 127:113-136
[10] Bauer W, Coburn L, Isralowitz J. Heat flow, BMO, and compactness of Toeplitz operators. J Funct Anal, 2010, 259:57-78
[11] Bauer W, Isralowitz J. Compactness characterization of operators in the Toeplitz algebra of the Fock space F_α^p. J Funct Anal, 2012, 263:1323-1355
[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
[13] Berezin F. Wick and anti-Wick symbols of operators (Russian). Mat Sb, 1971, 86:578-610
[14] Berezin F. Covariant and contra-variant symbols of operators. Math USSR-Izv, 1972, 6:1117-1151
[15] Berezin F. Quantization. Math USSR-Izv, 1974, 8:1109-1163
[16] Berezin F. Quantization of complex symmetric spaces (Russian). Izv Akad Ser Mat, 1975, 39:363-402
[17] Berezin F. General concept of quantization. Comm Math Phys, 1975, 40:153-174
[18] Berezin F. Introduction to Superanalysis. Dordrecht:Reidel, 1987
[19] Berger C, Coburn L. Toeplitz operators and quantum mechanics. J Funct Anal, 1986, 68:273-299
[20] Berger C, Coburn L. Toeplitz operators on the Segal-Bargmann space. Trans Amer Math Soc, 1987, 301:813-829
[21] Berger C, Coburn L. Heat flow and Berezin-Toeplitz estimates. Amer J Math, 1994, 116:563-590
[22] Berger C, Coburn L, Zhu K. BMO in the Bergman metric on the classical domains. Bull Amer Math Soc, 1987, 17:133-136
[23] Berger C, Coburn L, Zhu K. Function theory on Cartan domains and Berezin-Toeplitz symbol calculus. Amer J Math, 1988, 110:921-953
[24] Bommier-Hato H. Lipschitz estimates for the Berezin transform. J Funct Spaces Appl, 2010, 8:103-128
[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
[27] H. Cho, J. Park, and K. Zhu, Products of Toeplitz operators on the Fock space. Proc Amer Math Soc, 2014, 142:2483-2489
[28] Coburn L. A Lipschitz estimate for Berezin's operator calculus. Proc Amer Math Soc, 2005, 133:127-131
[29] Coburn L. Sharp Berezin-Lipschitz estimates. Proc Amer Math Soc, 2007, 135:1163-1168
[30] Coburn L. Berezin-Toeplitz quantization//Algebraic Methods in Operator Theory. Boston:Birkhauser, 1994:101-108
[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
[32] Coburn L, Li B. Directional derivative estimates for Berezin's operator calculus. Proc Amer Math Soc, 2008, 136:641-649
[33] Cuckovic Z, Li B. Berezin Transform, Mellin Transform and Toeplitz Operators. Complex Anal Oper Theory, 2012, 6:189-218
[34] Davidson K, Douglas R. The generalized Berezin transform and commutator ideals. Pacific J Math, 2005, 222:29-56
[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
[37] Englis M. Toeplitz operators and the Berezin transform on H². Linear Alg Appl, 1995, 223/224:171-204
[38] Englis M. Berezin transform and the Laplace-Beltrami operator. Algebra i Analiz, 1995, 7:176-195
[39] Englis M. Asymptotics of the Berezin transform and quantization on planar domains. Duke Math J, 1995, 79:57-76
[40] Englis M. Berezin quantization and reproducing kernels on complex domains. Trans Amer Math Soc, 1996, 348:411-479
[41] Englis M. Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integral Equations Operator Theory, 1999, 33:426-455
[42] Englis M, Otáhalová R. Covariant derivatives of the Berezin transform. Trans Amer Math Soc, 2011, 363:5111-5129
[43] Englis M, Zhang G. On the derivatives of the Berezin transform. Proc Amer Math Soc, 2006, 134:2285- 2294
[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
[46] Hedenmalm H, Korenblum B, Zhu K. Theory of Bergman Spaces. New York:Springer-Verlag, 2000
[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
[49] Janson S, Peetre J, Rochberg R. Hankel forms and the Fock space. Revista Mat Ibero-Amer, 1987, 3:58-80
[50] Karabegov A, Schlichenmaier M. Identification of Berezin-Toeplitz deformation quantization. J Reine Angew Math, 2001, 540:49-76
[51] Kilic S. The Berezin symbol and multipliers of functional Hilbert spaces. Proc Amer Math Soc, 1995, 123:3687-3691
[52] Korenblum B, Zhu K H. An application of Tauberian theorems to Toeplitz operators. J Operator Theory, 1995, 33:353-361
[53] Lee J. Properties of the Berezin transform of bounded functions. Bull Austral Math Soc, 1999, 59:21-31
[54] Li B. The Berezin transform and Laplace-Beltrami operator. J Math Anal Appl, 2007, 327:1155-1166
[55] Li B. The Berezin transform and m-th order Bergman metric. Trans Amer Math Soc, 2011, 363:3031-3056
[56] Luecking D, Zhu K. Composition operators belonging to the Schatten ideals. Amer J Math, 1992, 114:1127-1145
[57] Ma X, Marinescu G. Berezin-Toeplitz quantization on Kähler manifolds. J Reine Angew Math, 2012, 662:1-56
[58] Ma P, Yan F, Zheng D, Zhu K. Products of Hankel operators on the Fock space. J Funct Anal, 2019, 277:2644-2663
[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
[60] MacCluer B. Compact composition operators on H^p(B_n). Mich Math J, 1985, 32:237-248
[61] MacCluer B, Shapiro J. Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Canadian Math J, 1986, 38:878-906
[62] Nam K, Zheng D, Zhong C. m-Berezin transform and compact operators. Rev Mat Iberoam, 2006, 22:867-892
[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
[65] Peetre J. The Berezin transform and Ha-plitz operators. J Operator Theory, 1990, 24:165-186
[66] Rao N V. The range of the Berezin transform. J Math Sci (NY), 2018, 228(6):684-694
[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
[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
[70] Shklyarov D, Zhang G. The Berezin transform on the quantum unit ball. J Math Physics, 2003, 44(4344):4344-4373
[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
[72] Stroethoff K, Zheng D. Toeplitz and Hankel operators on Bergman spaces. Trans Amer Math Soc, 1992, 329:773-794
[73] Stroethoff K, Zheng D. Products of Hankel and Toeplitz operators on the Bergman space. J Funct Anal, 1999, 169:289-313
[74] Suarez D. Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev Mat Iberoam, 2004, 20:563-610
[75] Suarez D. Approximation and the n-Berezin transform of operators on the Bergman space. J Reine Angew Math, 2005, 581:175-192
[76] Suarez D. The essential norm of operators in the Toeplitz algebra on A^p(B_n). Indiana Univ Math J, 2007, 56:2185-2232
[77] Untenberger A, Upmeier H. The Berezin transform and invariant differential operators. Comm Math Phys, 1994, 164:563-597
[78] Zhang G. Berezin transform on compact Hermitian symmetric spaces. Manuscripta Math, 1998, 97:371- 388
[79] Zheng D. Hankel operators and Toeplitz operators on the Bergman space. J Funct Anal, 1989, 83:98-120
[80] Zheng D. The distribution function inequality and products of Toeplitz operators and Hankel operators. J Funct Anal, 1996, 138:477-501
[81] Zhu K. VMO, ESV, and Toeplitz operators on the Bergman space. Trans Amer Math Soc, 1987, 302:617-646
[82] Zhu K. Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J Operator Theory, 1988, 20:329-357
[83] Zhu K. Schatten class Hankel operators on the Bergman space of the unit ball. Amer J Math, 1991, 113:147-167
[84] Zhu K. Operator Theory in Function Spaces. American Mathematical Society, 2007
[85] Zhu K. Analysis on Fock Spaces. New York:Springer, 2012
[86] Zorboska N. The Berezin transform and radial operators. Proc Amer Math Soc, 2002, 131:793-800

Related Articles 15

[1]	Qixiang YANG, Haibo YANG, Chitin HON, Tao QIAN. $L^P$ BOUNDEDNESS OF COMMUTATOR AND HÖRMANDER CONDITION [J]. Acta mathematica scientia,Series B, 2025, 45(5): 2171-2189.
[2]	Ali ABKAR. APPROXIMATION IN WEIGHTED HOLOMORPHIC BESOV SPACES ON POLYDISK AND UNIT BALL [J]. Acta mathematica scientia,Series B, 2025, 45(5): 2208-2216.
[3]	Yongjiang DUAN, Sawlet JUNIS, Na ZHAN. BERGMAN PROJECTION AND TOEPLITZ OPERATORS ON WEIGHTED HARMONIC BERGMAN SPACES INDUCED BY DOUBLING WEIGHTS [J]. Acta mathematica scientia,Series B, 2025, 45(4): 1497-1513.
[4]	Chunxu XU. BOUNDED, COMPACT AND SCHATTEN CLASSES HANKEL OPERATORS ON WEIGHTED FOCK SPACES [J]. Acta mathematica scientia,Series B, 2025, 45(4): 1529-1554.
[5]	Chunyu DONG, Xin GUO, Qijian KANG. COMPACT LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON THE UNIT BALL [J]. Acta mathematica scientia,Series B, 2025, 45(3): 809-824.
[6]	Min HE, Maofa WANG, Jiale CHEN. GENERALIZED COUNTING FUNCTIONS AND COMPOSITION OPERATORS ON WEIGHTED BERGMAN SPACES OF DIRICHLET SERIES [J]. Acta mathematica scientia,Series B, 2025, 45(2): 291-309.
[7]	Ziyao LIU, Dashan FAN. CERTAIN OSCILLATING OPERATORS ON HERZ-TYPE HARDY SPACES [J]. Acta mathematica scientia,Series B, 2025, 45(2): 310-326.
[8]	Lixia Feng, Yan Li, Zhiyu Wang, Liankuo Zhao. TOEPLITZ OPERATORS BETWEEN WEIGHTED BERGMAN SPACES OVER THE HALF-PLANE^* [J]. Acta mathematica scientia,Series B, 2024, 44(5): 1707-1720.
[9]	Zuoling LIU, Hasi WULAN. PRODUCT TYPE OPERATORS ACTING BETWEEN WEIGHTED BERGMAN SPACES AND BLOCH TYPE SPACES [J]. Acta mathematica scientia,Series B, 2024, 44(4): 1327-1336.
[10]	Boyong Chen, Liangying Jiang. BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS [J]. Acta mathematica scientia,Series B, 2024, 44(3): 789-809.
[11]	Yong CHEN, Young Joo LEE. SUMS OF DUAL TOEPLITZ PRODUCTS ON THE ORTHOGONAL COMPLEMENTS OF FOCK-SOBOLEV SPACES [J]. Acta mathematica scientia,Series B, 2024, 44(3): 810-822.
[12]	Jian CHEN. ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES [J]. Acta mathematica scientia,Series B, 2024, 44(2): 431-444.
[13]	Xiaosheng LIN, Dachun YANG, Sibei YANG, Wen YUAN. MAXIMAL FUNCTION CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BOTH NON-NEGATIVE SELF-ADJOINT OPERATORS SATISFYING GAUSSIAN ESTIMATES AND BALL QUASI-BANACH FUNCTION SPACES [J]. Acta mathematica scientia,Series B, 2024, 44(2): 484-514.
[14]	Wei QU, Tao QIAN, Ieng Tak LEONG, Pengtao LI. THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS [J]. Acta mathematica scientia,Series B, 2024, 44(2): 567-582.
[15]	Wei Ding, Yan Tang, Yueping Zhu. THE BOUNDEDNESS OF OPERATORS ON WEIGHTED MULTI-PARAMETER LOCAL HARDY SPACES^* [J]. Acta mathematica scientia,Series B, 2024, 44(1): 386-404.