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Localising sets for sigma-algebras and related point transformations. (English) Zbl 0737.47032
As a means for the investigation of weighted composition operators (i.e. linear transformations $$W$$ on $$L^ 2$$ of a complete $$\sigma$$-finite measure space $$(X,\Sigma,m)$$ defined by $$Wf:=\phi f\circ T$$, where $$T$$ is a $$\Sigma$$-measurable mapping from $$X$$ onto $$X$$ such that $$m\circ T^{- 1}\ll m$$ and $$\phi$$ is a nonnegative measurable function on $$X$$) the conception of localizing sets is developed. A set $$A\in\Sigma$$ is said to localize a subalgebra $${\mathcal A}$$ of $$\Sigma$$, if for each $$B\in\Sigma$$ contained in $$A$$ there exists $$S\in{\mathcal A}$$ such that $$S\cap A=B$$. If $$(X,\Sigma,m)$$ is completely atomic the localizing sets for $$T^{- 1}\Sigma$$ ($$T$$ as above) can be characterized completely as those sets where $$T$$ is one-to-one. In general case $$A$$ with $$m(A)>0$$ localizes $$T^{-1}\Sigma$$ if and only if there is a nonnegative bounded function $$\phi$$ on $$A$$ with support $$A$$ such that the referring operator $$WW^*$$ is a multiplication operator. Some properties of $$W$$ as to be normal or centered have simple correspondences in this frame.
Reviewer: G.Garske (Hagen)

##### MSC:
 47B38 Linear operators on function spaces (general) 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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