Description of invariant subspaces of \(L^ p(\mu)\) by multiplication operators. (English) Zbl 0591.47006

Rezola, M. L.

Collect. Math. 35, 207-219 (1984).

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.)

Keywords:

invariant subspaces; multiplication operator; shift operators; conditional expectation operator