The Berezin transform and operators on spaces of analytic functions
Banach Center Publications (1997)
- Volume: 38, Issue: 1, page 361-380
- ISSN: 0137-6934
Access Full Article
topAbstract
topHow to cite
topStroethoff, Karel. "The Berezin transform and operators on spaces of analytic functions." Banach Center Publications 38.1 (1997): 361-380. <http://eudml.org/doc/208641>.
@article{Stroethoff1997,
abstract = {In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions 2. Definition and properties Berezin transform 3. Berezin transform and non-compact operators 4. Commutativity of Toeplitz operators 5. Berezin transform and Hankel or Toeplitz operators 6. Sarason's example},
author = {Stroethoff, Karel},
journal = {Banach Center Publications},
keywords = {commutativity of Toeplitz operators; Berezin transform; symbol; spaces of analytic functions; Hardy space; Bergman space; Fock space},
language = {eng},
number = {1},
pages = {361-380},
title = {The Berezin transform and operators on spaces of analytic functions},
url = {http://eudml.org/doc/208641},
volume = {38},
year = {1997},
}
TY - JOUR
AU - Stroethoff, Karel
TI - The Berezin transform and operators on spaces of analytic functions
JO - Banach Center Publications
PY - 1997
VL - 38
IS - 1
SP - 361
EP - 380
AB - In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions 2. Definition and properties Berezin transform 3. Berezin transform and non-compact operators 4. Commutativity of Toeplitz operators 5. Berezin transform and Hankel or Toeplitz operators 6. Sarason's example
LA - eng
KW - commutativity of Toeplitz operators; Berezin transform; symbol; spaces of analytic functions; Hardy space; Bergman space; Fock space
UR - http://eudml.org/doc/208641
ER -
References
top- [1] P. Ahern, M. Flores and W. Rudin, An invariant volume-mean-value property, J. Funct. Anal. 111 (1993), 380-397. Zbl0771.32006
- [2] J. Arazy, S. Fisher and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989-1054. Zbl0669.47017
- [3] S. Axler, Bergman spaces and their operators, in: Surveys of Some Recent Results in Operator Theory, Vol. I, J. B. Conway and B. B. % Morrell (eds.), Pitman Res. Notes, 1988, 1-50.
- [4] S. Axler, Berezin symbols and non-compact operators, unpublished manuscript, 1988.
- [5] S. Axler and Ž. Čučković, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory 14 (1991), 1-12. Zbl0733.47027
- [6] F. A. Berezin, Covariant and contravariant symbols of operators, Math. USSR-Izv. 6 (1972), 1117-1151. Zbl0259.47004
- [7] D. Békollé, C. A. Berger, L. A. Coburn and K. H. Zhu, BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93 (1990), 310-350. Zbl0765.32005
- [8] C. A. Berger and L. A. Coburn, Toeplitz operators and quantum mechanics, ibid. 68 (1986), 273-299.
- [9] C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), 813-829. Zbl0625.47019
- [10] C. A. Berger, L. A. Coburn and K. H. Zhu, Function theory on Cartan domains and Berezin-Toeplitz symbol calculus, Amer. J. Math. 110 (1988), 921-953. Zbl0657.32001
- [11] A. Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963), 89-102. Zbl0116.32501
- [12] J. A. Cima, K. Stroethoff and K. Yale, Bourgain algebras on the unit disk, Pacific J. Math. 160 (1993), 27-41. Zbl0816.46046
- [13] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. Zbl0469.30024
- [14] V. Guillemin, Toeplitz operators in n-dimensions, Integral Equations Operator Theory 7 (1984), 145-205. Zbl0561.47025
- [15] G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949. Zbl0032.05801
- [16] B. Korenblum and K. H. Zhu, An application of Tauberian theorems to Toeplitz operators, J. Operator Theory 33 (1995), 353-361. Zbl0837.47022
- [17] J. Peetre, The Berezin transform and Ha-plitz operators, J. Operator Theory 24 (1990), 165-186. Zbl0793.47026
- [18] P. Rosenthal, Berezin symbols and compactness of operators, unpublished manuscript, 1986.
- [19] W. Rudin, Function Theory in the Unit Ball of , Springer, New York, 1980. Zbl0495.32001
- [20] D. Sarason, personal communication.
- [21] J.H. Shapiro, The essential norm of a composition operator, Ann. of Math. 12 (1987), 375-404. Zbl0642.47027
- [22] K. Stroethoff, Compact Hankel operators on the Bergman space, Illinois J. Math. 34 (1990), 159-174. Zbl0687.47019
- [23] K. Stroethoff, Compact Hankel operators on the Bergman spaces of the unit ball and polydisk in , J. Operator Theory 23 (1990), 153-170. Zbl0723.47018
- [24] K. Stroethoff, Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39 (1992), 3-16. Zbl0774.47012
- [25] K. Stroethoff, Essentially commuting Toeplitz operators with harmonic symbols, Canad. Math. J. 45 (1993), 1080-1093. Zbl0803.47029
- [26] K. Stroethoff and D. Zheng, Toeplitz and Hankel operators on Bergman spaces, Trans. Amer. Math. Soc. 329 (1992), 773-794. Zbl0755.47020
- [27] K. H. Zhu, VMO, ESV, and Toeplitz operators on the Bergman space, ibid. 302 (1987), 617-646.
- [28] K. H. Zhu, Operator Theory in Function Spaces, Dekker, New York, 1990.
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.