Limited Set And Limited Operator In Banach Space

On the basis of collecting and rearranging most of the previous results , we study the limited operator and relavent problems in the view of operator ideals and space ideals in the sense of Pietsh. In addition, we also give new proof to several existing results with new method or from new angle. Section 1 and section 2 give the main results about the limited set and the limited operator. Further more,we prove that the set of all the limited operators LCD is a real and closed surjective operator ideal and that (L, LO) = LO, where L is the set of all the bounded linear operators. Section 3 mainly points it out that GP is a space ideal. In section 4 ,we use the method by which we discuss the limited set and the limited operator to discuss the weakly limited set and the weakly limited operator, which are respectively weaker than the limited set and the limited operator. And we also prove that the set of all the weakly limited operators WUNO is a closed surjective operator ideal and that (L, WUNO) = WUNO. In section 5, we study the set of all the limited operators LO(E, F) as a Banach space . Generalizing Kalton's method from the set of all the compact operators K(E, F) to LO(E,F), we get some parallel results.