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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.
##### MSC:
 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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