×

zbMATH — the first resource for mathematics

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.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fogel, The Ergodic Theory of Markov Processes (1969)
[2] DOI: 10.1090/S0002-9939-1978-0492057-5
[3] Morrell, Studia Math. 51 pp 251– (1974)
[4] DOI: 10.1017/S0004972700008479 · Zbl 0385.47017
[5] Halmos, A Hilbert Space Problem Book (1967)
[6] Nordgren, Hilbert Space Operators 639 (1978)
[7] Harrington, J. Operator Theory 11 pp 125– (1984)
[8] Hoover, Studio Math. LXXII pp 225– (1982)
[9] DOI: 10.1112/blms/18.4.395 · Zbl 0624.47014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.