Triangularizing semigroups of positive operators on an atomic normed Rieszspace

In the first part of the paper we prove several results on the existence of invariant closed ideals for semigroups of bounded operators on a normed Ri esz space (of dimension greater than 1) possessing an atom. For instance, i f S is a multiplicative semigroup of positive operators on such space that are locally quasinilpotent at the same atom, then S has a non-trivial invar iant closed ideal. Furthermore, if T is a non-zero positive operator that i s quasinilpotent at an atom and if S is a multiplicative semigroup of posit ive operators such that TS less than or equal to ST for all S is an element of S, then S and T have a common non-trivial invariant closed ideal. We al so give a simple example of a quasinilpotent compact positive operator on t he Banach lattice l(infinity) with no non-trivial invariant band. The second part is devoted to the triangularizability of collections of ope rators on an atomic normed Riesz space L. For a semigroup S of quasinilpote nt, order continuous, positive, bounded operators on L we determine a chain of invariant closed bands. If, in addition, L has order continuous norm, t hen this chain is maximal in the lattice of all closed subspaces of L.