The Study Of Stability (omega) Properties

The study of operator spectrum theory has been a significant subject and pop-ular branch of operator theory. In recent decades, with the rapid development of operator theory, people have focused on it, especially operator spectrum theory which not only has major applications in modern science and technology, quantum mechanics and modern physics, but also takes a direct effect on modern mathemat-ics, nonlinear science, computational mathematics and so on, such as the eigenvalue problem of differential equation, inverse scattering theory, signal analysis, ergodic theory. Perturbation theory of linear operator is closely linked with the disciplines of physics, engineering, quantum mechanics, for instance, spectrum distribution is referred to compute vibration frequency and determine the stability of system. Therefore, it, especially the perturbation of Weyl type theorems related to the dis-tribution of eigenvalue in quantum mechanics, has been an impressive key branch in operator theory.The main content of the paper is property (ω) which is a new variant of Weyl’s theorem. Mrs Cao has already proposed the definitions of consistency in invertibility and consistency in Fredholm and index, and has found they have close contacts with property (ω). Then they are used to property (ω) and its perturbation in this paper. Further, the perturbation theorem of property (ω) is put forward.This paper contains four chapters:In Chapter1, through defined spectrum, the judgment theorem and pertur-bation theorem of a bounded linear operator which has property (ω) are given. Meanwhile, we study the stability of algebraically quasi-class A operators by means of main conclusions.In Chapter2, perturbation theorem of operator which satisfies property (ω) is posed by the spectrum σCFI(·).In addition, we make a research on perturbation theorem of property (ω) of H(P) operators.In Chapter3, we establish the sufficient and necessary conditions and its per-turbation theorem for a bounded linear operator generalized property (ω) holds for. Then the stability of operators which can be decomposed the product of finite normal operators is discussed.In Chapter4, the perturbation theorem of a bounded linear operator which has property (ω) is given by way of topological uniform descent. In addition, the stability of algebraically paranormal operators which satisfy property (ω) is discussed.