The Research Of The Nature Of The Special Operator And Its Inverse In Riesz Space

The operator property is a famous problem in the history of operator theory. This paper studies the nature of some kinds of special operators and their inverse of operators in the Riese space, mainly it has three parts:In the first part, we investigate the nature of the special operators in the Riese space. First, based on the outcome of the preceding research, it discusses the nature of the disjointness preserving operator. Then, the nature of the band operator and studies the relationship between band operator and disjointness preserving operator. at last, discussing some equal problems for some conditions which have very close relations with the disjointness preserving operator.The second part is devoted to the nature of the inverse of operator for the disjointness preserving operator and so on. It discusses that the inverse of operator is still of this kind of nature under what kind of conditions. Similarly, we have obtained the result that inverse of operator must not be band operator under general condition by studying the nature of inverse of operator of band operator. At the same time providing some weak conditions which make inverse of operator is still band operator.Finally, we researched the regularity of special operators such as band operator and so on. first of all, it gives an example that explains not all of band operators are regular, secondly, it discusses that under what certain condition band operator is regular, at last it gives us that band operator is regular operator indeed.