Almost Dunford-Pettis Operators On Banach Lattices

The almost Dunford-Pettis operator is a new type of operators on Banach lattices, and play an important role in operator theory. Almost Dunford-Pettis operators and Dunford-Pettis operators are close connection. Many results about the Dunford-Pettis operator are obtained. By contrast, the properties of the almost Dunford-Pettis operators are scarce. This paper focus on the almost Dunford-Pettis operators. We discuss the controllability, conjugacy, M-weak and L-weak compactness of the almost Dunford-Pettis operators.In section one, we discuss the controllability and conjugacy of the almost Dunford-Pettis operators. The properties of the set constructed by all almost Dunford-Pettis operators are given. It is proved that the set of all almost Dunford-Pettis operators from E into F is a norm closed vector subspace in the vector space of all continuous operators from E into F.In section two, we mainly research the M-weak and L-weak compactness of the almost Dunford-Pettis operators. Firstly, it is obtained that each positive almost Dunford-Pettis operator from E into F is M-weakly compact if and only if the norm of E' is order continuous. Secondly, we discuss the L-weak compactness of the almost Dunford-Pettis operators. It is proved that if every positive almost Dunford-Pettis operator from E into a non-zero Banach lattice F is L-weakly compact, then the norm of F is order continuous. In addition, we give a necessary and sufficient condition for which each positive almost Dunford-Pettis operator from E into F is L-weakly compact if and only if F is finite-dimensional or the norm of E' and F are order continuous.In section three, some important conclusions are given about the b-weak and weak compactness of the almost Dunford-Pettis operators. It is proved that each almost Dunford-Pettis operator from E into F is weakly compact if and only if the norm of E' is order continuous or F is reflexive.