Research On Weyl Theorem And A-Weyl Theorem Of Operator Functions

As one of the most fundamental theories of modern mathematics,Operator spectrum theory is a enduring research subject of functional analysis.Operator spectrum theory has good applications in partial differential equation,nonlinear science and quantum mechanics.In addition,it has important and indispensable effect on modern physics and modern science and technology.H.Weyl proposed Weyl's theorem when he check up the spectrum of Hermitian operator.Then Weyl,s theorem was generalized by many scholars,so it occurred a-Weyl's t.heorem.In recent decades,Weyl's theorem and a-Weyl's theorem have been one of the hotpot directions of operator spectrum theory.It has important theoretical value and application value in the study of operator theory.On the basis of existing theory,Using the new spectrum,this paper characterize those operators T on H satisfying that a-Weyl's theorem holds for f(T)for each function f analytic on some neighborhood of ?(T),Meanwhile,we discussed those subcyclic operators T on H satisfying that Weyl's theorem holds for f(T)for each function f analytic on some neighborhood of ?(T).The paper contains three chapters,more details is as follows:In Chapter 1,we introduce the research background and fundamental knowledge,we.also give the defmitons of Wcyl's theorem,a-Weyl's theorem and the new spectrum.In Chapter 2,we use the new spectrum and discuss the determination method of a-Weyl's theorem for functions of operators,Meanwhile,we investigate the stability of the a-Weyl's theorem.In Chapter 3,we use the new spectrum and discuss the efficient and necessary conditions of subcyclic operators satisfying Weyl's theorem,including the functions of operators.At the same time,we also to illustrate the essence of conditions in the theorem.