Domination Property And Invariant Subspaces For AM-compact And Dunford-Pettis Operator

The invariant subspace problem is a famous problem in the history of Functional Analysis. This paper is denoted to the invariant subspaces of positive operators on banach lattice,such as AM-compact and Dunford-Pettis operator .Actually, the present work consists of two main parts. Main results are included in the first part,where we discuss the properties of AM-compact operator and its invariant subspaces. Based on the knowledge on AM-compactness at the past, we summarize its properties totally,in which its order structure is mainly studied , and get AM-compact operator's domination property. If a positive operator on Banach lattice is dominated by an AM-compact op-erator,then its square is also an AM-compact operator. Also, we introduce the concept of AM-compact-friendly operator and get a class of its invariant subspaces which generalized the relative result of compact-friendly operators.In the second part,we show that the Dunford-Pettis operator has also similar results of invariant subspace. For Dunford-Pettis operator has closed relation with AM-compact operator, we introduce the similar concept of Dunford-Pettis-friendlyness and get some relative results.