Irreducible operators

In this work; we give the classes of positive operators on a Banach lattice according to their special properties. These are the ideals and bands in terms of certain invariant subspaces.We presented conditions that guarantee the positivity of the spectral radius for compact positive operators. In particularly it is shown that if a compact quasi-nilpotent positive operator on a Banach lattice satisfies , then there exists a non-trivial closed lattice- hyperinvarient ideal J for T such that . The analogue of this result for lattice hyperinvarient bands was also given.Also, it was discussed the spectral radius of band irreducible positive operators. It was investigated band irreducible positive operators under the assumption ? -order continuous property.Keywords: Invariant subspaces, band, ideal, band irreducible, quasi-interior point, expanding operator, spectral radius, quasi-nilpotent.