| As one of the important topics in operator theory,operator spectrum theory has developed rapidly in recent decades,and has a wide range of applications in many mathematics and physics areas.The study of the spectral theory of normal opera-tors enables people to obtain deep understanding of the internal structure of normal operator spectrum.In operator theory,one of the most fundamental work is to generalize the theory of normal operator,and the local spectral theory is one of the important generalization.What’s more,the single valued extension property plays an important role in studying the local spectral theory.Since the concept of the single valued extension property has been put forward,it has been widely concerned by many mathematicians,and lots of scholars have been made great progress about the study of the single valued extension property at home and abroad.Meanwhile,operator matrices as a powerful tool to study the operator spectral theory,gradually become the focus of attention.Scholars have been made great research about spec-trum and some property of operator matrices.Let H be a complex separable infinite dimensional Hilbert space,A ∈ B(H),B ∈ B(K),on the basis of existing theory,this paper mainly discussed the stability of the single valued extension property for upper triangular operator matrix that on H(?)K under compact perturbation and small compact perturbation,then give some examples to prove the essence of the given conditions.This paper contains three chapters,and the specific arrangement is as follows:In the first chapter,we introduce the relevant background knowledge and vari-ous kinds of operators,and the definition of various spectral sets,at the same time,the related knowledge of the concept of the single value extension property and the upper triangular operator matrix is also given.In the second chapter,we defined a kind of spectrum,using the new spectrum to study the stability of the single valued extension property of upper triangular operator matrix that under compact perturbation,and gave the sufficient and nec-essary conditions of upper triangular operator matrix which satisfies the stability of the single valued extension property,then gave some examples to prove the essence of the given conditions.In the third chapter,by defining a kind of new resolvent set,we characterize upper triangular operator matrices for which satisfies the single valued extension property of small compact perturbations,at the same time,the sufficient and neces-sary conditions are given,and illustrate the conditions of upper triangular operator matrix which has stability are indispensable. |
