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Compact operators on the Bergman space of multiply-connected domains

Author: Roberto Raimondo
Journal: Proc. Amer. Math. Soc. 129 (2001), 739-747
MSC (2000): Primary 47B35
Published electronically: September 19, 2000
MathSciNet review: 1801999
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Abstract | References | Similar Articles | Additional Information


If $\Omega $ is a smoothly bounded multiply-connected domain in the complex plane and $A=\sum_{j=1}^m\prod_{k=1}^{m_j}T_{\varphi _{j,k}},$where $\varphi _{j,k}\in L^\infty ({\Omega },d{\nu }),$we show that $A$ is compact if and only if its Berezin transform vanishes at the boundary.

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  • [1] J. Arazy, Membership of Hankel Operators on Planar Domains in Unitary Ideals, Analysis at Urbana, vol.1, London Math. Soc. Lecture Notes Ser. 137, Cambridge University Press, 1989, 1-40. MR 90g:47048
  • [2] S. Axler and D. Zheng, Compact Operators via the Berezin Transform, Indiana Univ. Math. J. 47 (1998), 387-400. MR 99i:47045
  • [3] F. A. Berezin, Covariant and Contravariant Symbols of Operators, Math. USSR Izv. 36 (1972), 1117-1151. MR 50:2996
  • [4] V. Bergman, The Kernel Function and the Conformal Mapping, AMS Math. Surveys $\textbf{ 5},$ 1950.
  • [5] R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, 1972. MR 50:14335
  • [6] G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Trans. of Math. Monographs 26 Providence, R.I., 1969. MR 40:308
  • [7] L. Huiping, Hankel Operators on the Bergman Space of Multiply-Connected Domains, J. Oper. Theory 28 (1992), 321-335. MR 95d:47029
  • [8] N. Kerzman, The Bergman Kernel Function, Differentiability at the Boundary, Math. Ann. 195 (1972), 149-158. MR 45:3762
  • [9] B. Russo, On the Hausdorff-Young Theorem for Integral Operators, Pacific J. of Math. 68 (1977), 241-252. MR 58:17974

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Additional Information

Roberto Raimondo
Affiliation: Department of Economics, University of California at Berkeley, Evans Hall, Berkeley, California 94720

Received by editor(s): May 4, 1999
Published electronically: September 19, 2000
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2000 American Mathematical Society

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