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The Berezin transform and radial operators

Author: Nina Zorboska
Journal: Proc. Amer. Math. Soc. 131 (2003), 793-800
MSC (2000): Primary 47B37, 47B10; Secondary 47B35
Published electronically: July 2, 2002
MathSciNet review: 1937440
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Abstract | References | Similar Articles | Additional Information

Abstract: We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radial $L^{1}(\mathbb{D})$ symbol.

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  • 1. Ahern, P., Flore, M., and Rudin, W.: An Invariant Volume-Mean-Value Property, J. Funct. Anal. 111 (1993), 380-397. MR 94b:31002
  • 2. Axler, S., Zheng, D.: Compact Operators via the Berezin Transform, Indiana University Mathematics Journal 47 (1998), 387-400 MR 99i:47045
  • 3. Axler, S., Zheng, D.: The Berezin Transform on the Toeplitz Algebra, Studia Mathematica 127 (1998), 113-136 MR 98m:47030
  • 4. Englis, M.: Functions Invariant Under the Berezin Transform, J. Funct. Anal. 121 (1994), 223-254 MR 95h:31001
  • 5. Grudski, S., Vasilevski, N.: Bergman-Toeplitz Operators:Radial Component Influence, Integral Equations Operator Theory 40 (2001), no. 1, 16-33
  • 6. Kiliç, S.: The Berezin Symbol and Multipliers of Functional Hilbert Spaces, Proc. Amer. Math. Soc 123 (1995), 3687-3691 MR 96b:46035
  • 7. Korenblum, B. and Zhu, K. H.: An Application of Tauberian Theorems to Toeplitz Operators, Journal of Operator Theory 33 (1995), 353-361 MR 96i:47046
  • 8. Luecking, D.: Trace Ideal Criteria for Toeplitz Operators , Journal of Functional Analysis 73, (1987), 345-368 MR 88m:47046
  • 9. McDonald, G., Sundberg, C.:Toeplitz Operators on the Disk, Indiana University Math. J. 28 (1979), 595-611 MR 80h:47034
  • 10. Nordgren, E., Rosenthal, P.: Boundary Values of Berezin symbols, Operator Theory Advances and Applications 73 (1994), 362-368 MR 96b:46036
  • 11. Postnikov, A.G.: Tauberian Theory and its Applications, Proc. Steklov Inst. Math., Amer. Math. Soc., Vol. 144, 1980 MR 82f:40012b
  • 12. Stroethoff, K.: The Berezin Transform and Operators on Spaces of Analytic Functions, Banach Center Publ. 38 (1997), 361-380 MR 98g:47025
  • 13. Stroethoff, K.: Compact Toeplitz Operators on Bergman Spaces, Math. Proc. Cambridge Philos. Soc. 124 (1998), 151-160 MR 99i:47046
  • 14. Vukotic, D.: Carleson Measures and Bergman Spaces Operators, Preprint
  • 15. Zhu, K.: Positive Toeplitz Operators on Weighted Bergman Spaces of Bounded Symmetric Domains, J. Operator Theory 20 (1988), 329-357 MR 92f:47022
  • 16. Zhu, K.: Operator Theory in Function Spaces, Marcel Dekker, Inc., 1990 MR 92c:47031

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Additional Information

Nina Zorboska
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Keywords: Berezin transform, radial operator, Toeplitz operator
Received by editor(s): February 6, 2001
Received by editor(s) in revised form: October 12, 2001
Published electronically: July 2, 2002
Additional Notes: This work was supported by an NSERC grant
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

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