| As one of the important topics in operator theory, operator spectrum theory has developed rapidly in recent decades, and plays a very important role in many mathematics and physics. The study of the spectral theory of normal operators enables people to obtain deep understanding of the internal structure of normal operators. In linear operator theory, one of the most important work is to generalize the theory of normal operator. There is no doubt that local spectral theory is one of the important generalization. However, the single valued extension property is an important tool in the study of the local spectral theory, since the concept of the single valued extension property was put forward, the study of it have much more concerned, and a lot of scholars, at home and abroad, have been made great achievement about the study of the single valued extension property.In addition, with the development of linear operator theory, the study of oper-ator matrices has been the focus of attention gradually. Scholars have been made great research about spectrum and some property of the upper triangular operator matrix. Let H is a complex separable 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, here, C∈B(K,H).This paper contains three chapters, and the specific arrangement is as follows:In the first chapter, mainly introduce the relevant background and the definition of various operators and spectrum, at the same time, the concepts of the single valued extension property and so on are given.In the second chapter, we define new spectrum, using the new spectrum to study the stability of the single valued extension property of bounded and linear operator that under compact perturbation and small compact perturbation, then using the related theorem of the single valued extension property which has stability, gave the sufficient and necessary conditions of upper triangular operator matrix which has stability.In the third chapter, we study the upper triangular operator matrix Mc, mainly introduce the stability of upper triangular operator matrix of the single valued ex-tension property under compact perturbation and small compact perturbation with A and B that on the main diagonal of upper triangular operator matrix, at the same time, the sufficient and necessary conditions are given, and illustrate the conditions of upper triangular operator matrix which has stability are indispensable. |
