Invariant subspace problem for some spaces and classes of operators

This thesis has reached two main goals. The one of them is to give some results about the compactness property of L-weakly and M-weakly compact operators that are the subclasses of weak compact operators defined on Banach lattices. The other one is to investigate some circumtances that L-weakly compact and M-weakly compact operators having non-trivial closed invariant subspace. And finally, the obtained results have also helped us to show that some positive operators defined on Banach lattice have non-trivial closed invariant subspace.