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Description of invariant subspaces of \(L^ p(\mu)\) by multiplication operators. (English) Zbl 0591.47006
The main result is a description of the closed subspaces of \(L^ p(X,{\mathfrak A},\mu)\), \(1\leq p<\infty\), which are invariant under multiplication by a selfconjugate family H of essentially bounded functions. When the sub \(\sigma\)-algebra \(\sigma\) (H), generated by H, is \(\sigma\)-finite this description is obtained by using the conditional expectation operator relative to \(\sigma\) (H). If \(\sigma\) (H) is not \(\sigma\)-finite, but it is invariant by a group of transformations, we obtain a similar result with a suitable substitute of the conditional expectation operator. Moreover some examples and applications are given.
47A15 Invariant subspaces of linear operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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