Operator semigroups for which reducibility implies decomposability (Q1417875)

Let \({\mathcal S}\) be a multiplicative semigroup of bounded operators acting on \(L^p (X, \mu)\), where \(1 \leq p < \infty\) and \((X, \mu)\) is a \(\sigma\)-finite measure space. \(L^\infty (X, \mu)\) is identified with the space of multiplication operators \(M_\varphi\) given by \(M_\varphi f = \varphi f\), \(\varphi \in L^\infty (X, \mu)\). For certain semigroups \({\mathcal S}\), every invariant subspace is the direct sum of an invariant band with a subspace that is mapped into that band. More precisely, the authors prove the following Theorem: Suppose that the semigroup \({\mathcal S}\) has the property that \(L^p {\mathcal S} L^\infty \subset {\mathcal S}\). If \(M\) is an invariant subspace for \({\mathcal S}\), let \(L^p (Z)\) be the band generated by \(M\) and \(L^p (Y)\) be the band generated by \({\mathcal S} M\). Then \(L^p (Y) \subseteq M \subseteq L^p (Z)\) and \({\mathcal S} L^p (Z) \subseteq L^p (Y)\). Moreover, any subspace \(N\) with \(L^p (Y) \subseteq N \subseteq L^p (Z)\) is also invariant for \({\mathcal S}\) and can be written as \(N = L^p (Y) \oplus N_Y\), where \(N_Y\) is a subspace in \(L^p (Z/Y)\). Let \(\operatorname{lat}{\mathcal S}\) be the lattice of all invariant subspaces of \({\mathcal S}\). A subspace \(M\) of \(L^p\) is a band of \(L^p\) if \(M = L^p (E, \mu | _E)\) for some measurable subset \(E\) of \(X\). Let \(\operatorname{slat}{\mathcal S}\) denote the lattice of all invariant bands of \({\mathcal S}\). Theorem: Let \({\mathcal S}\) be a semigroup acting on \(L^p\) with \(L^\infty {\mathcal S} L^\infty \subseteq {\mathcal S}\), then \(\operatorname {lat}{\mathcal S}\) is distributive if and only if \(\operatorname {lat}{\mathcal S} = \operatorname {slat}{\mathcal S}\). These theorems have a number of interesting corollaries. The authors give simpler proofs of known results and applications to semigroups of positive quasinilpotent operators.