KB-OPERATORS ON BANACH LATTICES AND THEIR RELATIONSHIPS WITH DUNFORD-PETTIS AND ORDER WEAKLY COMPACT OPERATORS

Aqzzouz, Moussa and Hmichane proved that an operator T from a Banach lattice E into a Banach space X is b-weakly compact if and only if {T-x}(n) is norm convergent for every positive increasing sequence {x(n)}(n) of the closed unit ball B-E of E. In the present paper, we introduce and study new classes of operators that we call KB operators and W KB-operators. A continuous operator T from a Banach lattice E into a Banach space X is said to be KB-operator (respectively, W KB-operator) if {Tx(n)}(n) has a norm (respectively, weak) convergent subsequence in X for every positive increasing sequence {x(n)}(n) in the closed unit ball B-E of E. We investigate the relationships between KB-operators (respectively, W KB-operators) and some other operators on Banach lattices spacial their relationships with Dunford-Pettis and order weakly compact operators.