Invariant Subspaces and Spectral Conditions on Operator Semigroups

0. Introduction. Let H be a complex Hilbert space of finite or infinite dimension, and let E be a collection of bounded linear operators on H. We say E is reducible if there exists a subspace of H, closed by definition and different from the trivial subspaces {0} and H which is invariant under every member of E . We call E triangularizable if the set of invariant subspaces under E contains a maximal subspace chain. These questions have been studied extensively and the central problems in the infinite-dimensional case, i.e., the invariant subspace problem and the transitive algebra problem are still unsolved for arbitrary operators on H [20, 21]. We are interested in the effect of certain spectral conditions on reducibility and triangularizability of a collection E . If f is any function, defined at least on all products of members of E , we say that f is permutable if