Study On The Strongly Ireducible Operators On Banach

This paper studies on some special strongly irreducible operators and the existence of strongly irreducible operators on Banach spaces.There are three chapters in the paper.In Chapter 1 we introduce the background of the study of strongly irreducible operators,specify the prerequisites needed in the following chapters.Moreover, a detailed introduction is made on the strongly irreducible operators on Hilbert spaces.Chapter 2 discusses the existence of strongly irreducible operators on Banach spaces.We show that there are strongly irreducible operators on Banach space X whose conjugate space X'is w^* separable,and we generalize some results of strongly irreducible operators on Hilbert spaces to those on Banach spaces.Chapter 3 focuses on some special kinds of strongly irreducible operators,including those on Sobolev spaces W^2,p(Ω)(1＜p＜∞)and hereditarily indecomposable spaces,also including Cowen-Douglas operators with index 1 on Banach spaces.