Characterization of abstract composition operators

Abstract

A composition operator on L p ( X , μ ) {L^p}(X,\mu ) is (roughly) an operator T T induced by a point transformation ϕ \phi on X X by T f = f ⋅ ϕ 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 ) {L^2}(J,m) ( J J an interval of the real line, m m a Borel measure) and on L 2 ( 0 , 1 ) {L^2}(0,1) (Lebesgue measure) are considered. Also, any bounded measure-algebra transformation which preserves disjoint unions is a sigma-homomorphism.