Study Of Stability Of The Single Valued Extension Property On Operator Matrices

In linear operator theory, the local spectral theory has always been an important subject. As early as Weyl’s theorem was found in 1909, people began the study of Weyl’s theorem on operators. We know Browder’s theorem is the precondition of Weyl’s theorem, and the single valued extension property plays a very important role in Browder’s theorem. Therefore, since the concept of the single valued extension property was put forward, many scholars have started to study the single valued extension property of operators. In addition, if there is any bounded linear operator whose space can be decomposed into direct sum, this operator can be written to a operator matrix. As a result, in the field of linear operators, we often study* the nature of the operator itself with the concept of operator matrices.This paper probes into the stability of the single valued extension property and Browder’s theorem on a special kind of upper triangular operator matrices, and antidiagonal operator matrices under the tight perturbation. Through the study, the equivalent conditions of the single valued extension property and Browder’s theorem of the above two types of operator matrices under tight perturbation are concluded.The full text is divided into three chapters, and the specific arrangement is as follows:The first chapter is the introduction. On the one hand, the study and the related background knowledge is described, on the other hand, the definition of operators and the corresponding spectrums are given, in addition, the concepts and properties are explained.In the second chapter, we study a special kind of upper triangular operator matrices, discuss the stability of the single valued extension property and Browder’s theorem of the matrices under tight perturbation, depict the property of the oper-ator matrices with the related property of the first position operator, and then we conclude the equivalent conditions of the single valued extension property and Brow-der’s theorem of upper triangular operator matrices under all tight perturbations, with appropriate examples enumerated to illustrate this conclusion.In the third chapter, we study the stability of the single valued extension prop-erty and Browder’s theorem of the antidiagonal operator matrices under tight per-turbation. In addition, we conclude the equivalent conditions of the single valued extension property and Browder’s theorem of the antidiagonal operator matrices under all tight and tiny tight perturbations.