Limited Operators And Weakly Null Sequences On Banach Lattices

We investigate a new type of operators:almost weak* Dunford-Pettid operators. Some related characterizations of positive almost weak* Dunford-Pettid operators were showed. The so-called "unbounded norm convergence" is studied at last.With the defination of almost weak* Dunford-Pettid operators and the skills of constructing disjoint sequences, some characterizations of positive almost weak* Dunford-Pettid operators as well as their relationships with other operators are showed. In the same way, we investigate positive almost limited operators and positive limited operators. Then, we discusse their relations with Dunford-Pettis operators and M-weakly compact operators.Then, we investigate unbounded norm convergence in Banach lattices. It is worth to mention that we can always find a disjoint sequence to approximate a subsequence of an unbounded norm null sequence. Under different conditions, one can find a disjoint sequence to approximate a subsequence which can be founded from a positive weakly null sequence as well as a given weakly null sequence. We showed some relationships of these two types of convergences. Finally, we give another proof for the classic Diagoanl lemma.