The Relationship between M-Weakly Compact Operator and Order Weaky Compact Operator

In this note, we will show that the class of order weakly compact operators bigger than the class of M-weakly compact operators. Under a new condition, we will show that each M-weakly compact operator is an order weakly compact operator. We will show that, if Banach lattice E be an AM-space with unit and has the property (b), then the class of M-weakly compact operators from E into Banach space Y coincides with that of order weakly compact operators from E into Y. Also we establish some relationship between M-weakly compact operators and weakly compact operators and b-weakly compact operators and order weakly compact operators.