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Characterization of abstract composition operators


Author: William C. Ridge
Journal: Proc. Amer. Math. Soc. 45 (1974), 393-396
MSC: Primary 47B37
DOI: https://doi.org/10.1090/S0002-9939-1974-0346585-X
MathSciNet review: 0346585
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Abstract: A composition operator on $ {L^p}(X,\mu )$ is (roughly) an operator $ T$ induced by a point transformation $ \phi $ on $ X$ by $ Tf = f \cdot \phi $.

Characterizations are given of abstract Hilbert-space operators which can be represented (via unitary equivalence) as composition operators. Representation on $ {L^2}(J,m)$ ($ J$ an interval of the real line, $ m$ a Borel measure) and on $ {L^2}(0,1)$ (Lebesgue measure) are considered.

Also, any bounded measure-algebra transformation which preserves disjoint unions is a sigma-homomorphism.


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

  • [1] J. R. Choksi, Non-erodic transformations with discrete spectrum, Illinois J. Math. 9 (1965), 307-320. MR 30 #4903. MR 0174703 (30:4903)
  • [2] -, Unitary operators induced by measure-preserving transformations, J. Math. Mech. 16 (1966), 83-100. MR 34 #1844. MR 0201967 (34:1844)
  • [3] -, Unitary operators induced by measurable transformations, J. Math. Mech. 17 (1967/68), 785-801. MR 36 #2003. MR 0218919 (36:2003)
  • [4] W. C. Ridge, Composition operators, Thesis, Indiana University, Bloomington, Ind., 1969, pp. 1-11, 17-22.
  • [5] -, Spectrum of a composition operator, Proc. Amer. Math. Soc. 37 (1973), 121-127. MR 0306457 (46:5583)
  • [6] R. Sikorski, Boolean algebras, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 25, Academic Press, New York; Springer-Verlag, Berlin, 1964, p. 139. MR 31 #2178. MR 0177920 (31:2178)
  • [7] B. Sz.-Nagy, Über die Gesamtheit der charakteristischen Funktionen im Hilbertschen Funktionenraum, Acta Sci. Math. (Szeged) 9 (1937), 166-176.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0346585-X
Keywords: Linear operators, Hilbert space, measure algebra, measurable transformations, sigma-homomorphisms
Article copyright: © Copyright 1974 American Mathematical Society

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