The authors show that certain positive semigroups of quasinilpotent operators have invariant subspaces. In particular, a quasinilpotent operator having a matrix representation with non-negative entries has non-trivial invariant subspaces, and a similar result holds for integral operators with non-negative lower semicontinuous kernels. The results for a single operator are special cases of the Ando-Krieger theorem but the results here use an elementary approach and apply to semigroups. Results on triangularization of semigroups of operators, as mentioned above, are also given.