Reducible semigroups of idempotent operators

We study the existence of common invariant subspaces for semigroups of idem potent operators. It is known that in finite dimensions every such semigrou p is simultaneously triangularizable. The question; of the existence of eve n one non-trivial invariant subspace is still open in infinite dimensions. Working with semigroups of idempotent operators in Hilbert/Banach vector sp ace settings, we exploit the connection between the purely algebraic struct ure and the operator structure to show that the answer is affirmative in a number of cases.