Invariant Subspaces Families Of Positive Operators

Let be a collection of bounded linear operators on a Banach space X of dimension at least two. İt is known that is finitely quasinilpotent at a vektor whenever for any finite subset at is equal to 0. İf such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant subspace. İf, in addition, is a collection of positive operators on a Banach lattice, then 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 be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal. KEYWORDS: Positive operator, quasinilpotent, Banach lattice, compact operator, invariant subspace