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1997 | 38 | 1 | 361-380
Tytuł artykułu

The Berezin transform and operators on spaces of analytic functions

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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
Słowa kluczowe
Rocznik
Tom
38
Numer
1
Strony
361-380
Opis fizyczny
Daty
wydano
1997
Twórcy
  • Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812-1032, U.S.A.
Bibliografia
  • [1] P. Ahern, M. Flores and W. Rudin, An invariant volume-mean-value property, J. Funct. Anal. 111 (1993), 380-397.
  • [2] J. Arazy, S. Fisher and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989-1054.
  • [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  • [4] S. Axler, Berezin symbols and non-compact operators, unpublished manuscript, 1988.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.
  • [6] F. A. Berezin, Covariant and contravariant symbols of operators, Math. USSR-Izv. 6 (1972), 1117-1151.
  • [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.
  • [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.
  • [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.
  • [11] A. Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963), 89-102.
  • [12] J. A. Cima, K. Stroethoff and K. Yale, Bourgain algebras on the unit disk, Pacific J. Math. 160 (1993), 27-41.
  • [13] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
  • [14] V. Guillemin, Toeplitz operators in n-dimensions, Integral Equations Operator Theory 7 (1984), 145-205.
  • [15] G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949.
  • [16] B. Korenblum and K. H. Zhu, An application of Tauberian theorems to Toeplitz operators, J. Operator Theory 33 (1995), 353-361.
  • [17] J. Peetre, The Berezin transform and Ha-plitz operators, J. Operator Theory 24 (1990), 165-186.
  • [18] P. Rosenthal, Berezin symbols and compactness of operators, unpublished manuscript, 1986.
  • [19] W. Rudin, Function Theory in the Unit Ball of $ℂ^n$, Springer, New York, 1980.
  • [20] D. Sarason, personal communication.
  • [21] J.H. Shapiro, The essential norm of a composition operator, Ann. of Math. 12 (1987), 375-404.
  • [22] K. Stroethoff, Compact Hankel operators on the Bergman space, Illinois J. Math. 34 (1990), 159-174.
  • [23] K. Stroethoff, Compact Hankel operators on the Bergman spaces of the unit ball and polydisk in $C^n$, J. Operator Theory 23 (1990), 153-170.
  • [24] K. Stroethoff, Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39 (1992), 3-16.
  • [25] K. Stroethoff, Essentially commuting Toeplitz operators with harmonic symbols, Canad. Math. J. 45 (1993), 1080-1093.
  • [26] K. Stroethoff and D. Zheng, Toeplitz and Hankel operators on Bergman spaces, Trans. Amer. Math. Soc. 329 (1992), 773-794.
  • [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.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
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