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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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