Weyl-type Theorems For Bounded Linear Operators And Their Perturbations

Spectral theory of operators is an important research field of the theory of operators.Since many problems in physics,quantum mechanics,engineering and other disciplines can be transformed into solving operator equations(such as alge-braic equations,differential and integral equations,etc.)and the solutions to these problems are closely related to the eigenvalues of the operators.Thus,the study of spectral theory of operators is of great significance.For spectral theory of operators,the study of the spectral structures is a hot topic,which is helpful to solve operator equations and other related problems.In an infinite dimensional space,the spectral structure of an operator is relatively complex.According to whether the range of an operator is closed or not,and the finity or infinity of nullity and defect,the spectrum of the operator is classified into some spectral subsets,such as the semi-Fredholm spectrum,the Fredholm spectrum,the Weyl spectrum,etc..Scholars were interest-ed in the study of spectral structures of operators stimulated by these classifications.In 1909,H.Weyl found that the Weyl spectrum of a self-adjoint operator acting on a Hilbert space was precisely equal to the spectrum except for all Riesz points of the self-adjoint operator.This result was called Weyl's theorem by scholars and has aroused their attention to Weyl's theorem for operators and its related problems.In this paper,we define four different spectral subsets based on the semi-Fredholm theory and the local spectral theory.Using the spectral subsets,we give some e-quivalent characterizations,which are different from the traditional definitions.In addition,we also study the stabilities of Weyl's theorem under compact perturba-tions for positive integer power of operators,functional calculus for operators and operator matrices.Since the single valued extension property is an important tool for the study of spectral theory and is closely related to Weyl's theorem,we combine it with Weyl's theorem to study the spectral structures of operators.The following is a brief description of the main contents of this article.In Chapter 1,we first introduce the research background and the present sit-uations of Weyl theorem.Moreover,we give an account of the symbols and terms that will appear in the sequel.In addition,the main conclusions of this paper are briefly introduced.In Chapter 2,we study Weyl-type theorems for linear operators and their com-pact perturbations.First of all,several new spectral subsets are constructed by using the Weyl spectrum,the essential approximation point spectrum and the S-aphar spectrum.By means of the spectral subsets above,we give the judgements for an operator T satisfying the stabilities of Weyl-type theorems under compact perturbations.In addition,we study the stability of Weyl's theorem under compact perturbations for T3,and discuss the relations of the stability of Weyl's theorem between T and T3.In Chapter 3,we study Weyl's theorems for functional calculus for operators and their compact perturbations.Using the spectral subsets defined in Chapter 2,we first study the stability of Weyl's theorem for Tn(n?N+),and verify that under certain conditions,the stabilities of Weyl's theorem for T and Tn are equivalent.Using the spectral subsets again,we characterize the equivalent conditions that make functional calculus for operators satisfy Weyl's theorem and their stabilities.It is also proved that there are certain relations between the stability of Weyl's theorem for functional calculus for operators and the spectral mapping theorems for the newly defined spectral subsets.In Chapter 4,we study the stabilities of Weyl's theorem and the single valued extension property under compact perturbations for third-order upper triangular operator matrices.First of all,we characterize the equivalent conditions for oper-ator matrices satisfying the stability of Weyl's theorem.By research,we find that the stability of Weyl's theorem for a third-order upper triangular operator matrix is closely related to the operators on the diagonal.In addition,it is found that if a third-order upper triangular operator matrix satisfies the stability of single valued extension property,so do the operators on the diagonal.Finally,the relations be-tween the two stabilities are discussed,and it is proved that if the spectra of the diagonal operators are mutually disjoint,all of the operators on the diagonal satisfy the stabilities of Weyl's theorem and the single-valued extension property.