Study On Problems Of The Weyl Type Theorems

Spectral theory has always been a hot issue in the study of operator theory,and the more active study on spectral theory is Weyl type theorems in recent years.We make a deep study of Weyl type theorems for bounded linear operators on the Ba-nach space,and discuss a new class of operator—infinitely improjective operator,which is closely related to Riesz operators.This paper consists of four chapters.In chapter 1,the definitions of two new spectral properties are given.Then we discuss the relations between these two new properties and other Weyl type theorems.We also study their perturbations and direct sum results,and the new properties for Tn,the restriction of operator T.In chapter 2,we further study operators which admit a generalized Kato decompositon.And the characterizations of Browder theorem and Weyl theorem are discussed from the angle of generalized Kato spectrum.In chapter 3,we study the properties of(n,k)-quasi-*-paranormal operators.Then we discuss the Weyl type theorems for algebraically(totally)(n,k)-quasi-*-paranormal operators and spectral continuity.In chapter 4,the definition and properties of infinitely improjective operator are given.Then we discuss their links with other classes of operators,provide an intrinsic characterization of the inessential opera-tor,and characterize indecomposable spaces and the spaces satisfying scalar plus compact problem with infinitely improjective operators.