Properties Of Disjointness Preserving Operator On Classical Banach Lattices

Disjointness preserving operators are important special operators on Riesz space. This thesis is denoted to the properties of the disjointness preserving operator on classical Banach lattices,such as the range of disjointness preserving operator,the sum of disjointness preserving operator and so on.Actually,the present work consists of two main parts. In the first part,we discuss the problems of range space of disjointness preserving operator on classical Banach lattices. Based on other people's researches, We obtain the characterizations that the range spaces of disjointness preserving operators are Riesz subspace, ideal and band. Firstly, the necessary and sufficient condition for the range space of disjointness preserving operator T to be Riesz subspace is that the nonzero elements of matrix of T have the same symbol for every rows. Secondly, We give the matrix characterization of interval preserving operators, which is the necessary and sufficient condition for the range space of disjointness preserving operator T to be ideal. It is that the matrix of T satisfies that every row and every array have at most one nonzero element. And this is also the sufficient condition for the range spaces of disjointness preserving operators T to be band.In the second part, the properties of the sums of disjointness preserving operators on classical Banach lattices are discussed. the necessary and sufficient condition for the sum of two disjointness preserving operators to be disjointness preserving operator, is that their factors are linear relations, S_k =α_kT_k(k∈K).