Characterization of abstract composition operators

Ridge, William C.

doi:10.1090/S0002-9939-1974-0346585-X

Characterization of abstract composition operators
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Proc. Amer. Math. Soc. 45 (1974), 393-396 Request permission

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

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J. R. Choksi, Unitary operators induced by measurable transformations, J. Math. Mech. 17 (1967/1968), 785–801. MR 0218919

Composition operators

William C. Ridge, Spectrum of a composition operator, Proc. Amer. Math. Soc. 37 (1973), 121–127. MR 306457, DOI 10.1090/S0002-9939-1973-0306457-2
Roman Sikorski, Boolean algebras, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 25, Academic Press, Inc., New York; Springer-Verlag, Berlin-New York, 1964. MR 0177920

Über die Gesamtheit der charakteristischen Funktionen im Hilbertschen Funktionenraum

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