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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Characterization of abstract composition operators


Author: William C. Ridge
Journal: Proc. Amer. Math. Soc. 45 (1974), 393-396
MSC: Primary 47B37
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.


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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