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Proceedings of the American Mathematical Society

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Compact and quasinormal composition operators


Author: Raj Kishor Singh
Journal: Proc. Amer. Math. Soc. 45 (1974), 80-82
MSC: Primary 47B37
MathSciNet review: 0348545
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Abstract: Let $ {C_\phi }$ be a composition operator on $ {L^2}(\lambda )$, where $ \lambda $ is a $ \sigma $-finite measure on a set $ X$. If $ X$ is nonatomic, then Ridge proved that no one-to-one composition operator $ {C_\phi }$, with dense range is compact. This result is generalized in the paper by removing one-to-one and dense range conditions. The quasinormal composition operators are also characterized in terms of commutativity with the multiplication operator induced by the Radon-Nikodym derivative of the measure $ \lambda {\phi ^{ - 1}}$ with respect to $ \lambda $.


References [Enhancements On Off] (What's this?)

  • [1] W. C. Ridge, Composition operators, Thesis, Indiana University, 1969.
  • [2] R. K. Singh, Composition operators (to appear).
  • [3] Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Completely revised edition of An introduction to the theory of integration. MR 0222234

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0348545-1
Keywords: Composition operators, compact operators, quasinormal composition operators, atomic measures, nonatomic measures
Article copyright: © Copyright 1974 American Mathematical Society