LIMITED AND DUNFORD-PETTIS OPERATORS ON BANACH LATTICES

This paper is devoted to investigation of conditions on a pair of Banach lattices E; F under which every positive Dunford-Pettis operator T : E -> F is limited. Mainly, it is proved that if every positive Dunford-Pettis operator T : E -> F is limited, then the norm on E' is order continuous or F is finite dimensional. Also, it is proved that every positive Dunford-Pettis operator T : E -> F is limited, if one of the following statements is valid: (1) The norm on E' is order continuous, and F' has weak* sequentially continuous lattice operations. (2) The topological dual E' is discrete and its norm is order continuous. (3) The norm of E' is order continuous and the lattice operations in E' are weak* sequentially continuous. (4) The norms of E and of E' are order continuous.