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Integral Equations and Operator Theory

, Volume 18, Issue 2, pp 202–210 | Cite as

Fredholm composition operators on spaces of holomorphic functions

  • Osamu Hatori
Article

Abstract

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorCφ on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC2 boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if φ is a holomorphic automorphism.

MSC 1991

46J15 47A53 

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References

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    P. S. Bourdon,Fredholm multiplication and composition operators on the Hardy space, Integral Eq. and Operator Th.13 (1990), 607–610.Google Scholar
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    J. A. Cima, J. E. Thomson and W. R. Wogen,On some properties of composition operators, Indiana Univ. Math. J.24 (1974), 215–220.Google Scholar
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    R. M. Range,Holomorphic Functions and Integral Representations in Several Complex Variables, Springer, New York, Berlin, Haidelberg, Tokyo, 1986.Google Scholar
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    H. J. Schwartz,Composition operators on H p, Thesis, University of Toledo, 1969.Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • Osamu Hatori
    • 1
  1. 1.Department of MathematicsTokyo Medical CollegeTokyo 160Japan

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