Common invariant subspaces for collections

Let C be a collection of bounded operators on a Banach space X of dimension at least two. We say that C is finitely quasinilpotent at a vector x(0) is an element of X whenever for any finite subset F of C the joint spectral radius of F at x(0) is equal 0. If such collection C contains a non-zero compact operator, then C and its commutant C' have a common non-trivial invariant subspace. If, in addition, C is a collection of positive operators on a Banach lattice, then C has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let S be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then S has a common non-trivial invariant closed ideal.