| As one of the important topics in operator theory, operator spectrum theory has developed rapidly in recent decades. Operator spectrum theory not only has major applications in modern science and technology, quantum mechanics and mod-ern physics, but also takes a direct effect on modern mathematics, nonlinear science, computational mathematics and so on, such as the eigenvalue problem of differential equation, inverse scattering theory, signal analysis, ergodic theory. The study of the spectral theorem of normal operators has been more thorough and perfect. The spectral theorem of normal operators enables people to obtain a deep understand-ing of the internal structure of normal operators. In operator theory, one of the most fundamental objects is to generalize the theory of normal operators. There is no doubt that local spectrum theory is one of the important promotion. The single valued extension property is very useful in the study of the local spectra the-ory. In addition, perturbation theory of linear operator is closely linked with the disciplines of physics, engineering, quantum mechanics, for instance, spectrum dis-tribution is referred to compute vibration frequency and determine the stability of system. Therefore, it, especially the perturbation of Weyl type theorems related to the distribution of eigenvalue in quantum mechanics, has been an impressive key branch in operator theory.The main content of the paper is property (ω1) which is a new variant of Weyl’s theorem and the single valued extension property for the Helton class operators.This paper contains three chapters:In Chapter1, First we introduce the related background of the thesis writing and the basic symbol. Second the properties of the Helton class operators is given.In Chapter2, we investigate the stability of single valued extension property under compact perturbations for the Helton class operators. Also, we character-ize2x2upper triangular operator matrices for which the single valued extension property is stable under compact perturbations.In Chapter3, we investigate the stability of the property (ω1) under compact perturbations for the Helton class operators. In addition, we characterize2×2upper triangular operator matrices for which the property (ω1) is stable under compact perturbations. |
