On Related Problems Of Spectrum Of Linear Operators

The operator theory is a very important research field in functional analysis.It is a branch of mathematics that deeply reflects the essence of many mathematical problems,and has very important application value and deep research significance.The spectrum and related problems of linear operators are very important parts of operator theory,which are widely used in many branches of mathematics and physics.Such as matrix theory,differential equations,integral equations,control theory and quantum mechanics.Because a finite square matrix is either injective or singular,there is only one kind of spectral points,i.e.point spectrum.However,for a linear operator in an infinite dimensional space their spectra can be divided from different viewpoints into the point spectrum,residual spectrum,continuous spectrum,essential spectrum,discrete spectrum,etc.Therefore,the spectrum of linear operator is a generalization of matrix case.As the structure and types of spectra of a matrix are less and simpler than those of a linear operator,it is relatively easy to characterize its spectrum.But some types of spectra of a linear operator involve many properties associated with its range,null space,etc.So their characterizations are more difficult,and even impossible to describe in detail.Based on this,the main topic of this dissertation is related to the location of spectra of linear operators,and we focus on the important role of numerical range of linear operators in the characterization of spectra and the symmetry of the spectrum of symplectic self-adjoint Hamiltonian operators,which is expected to provide a theoretical basis for further discussion of spectral problems of linear operators.An important problem in the infinite dimensional Hamiltonian operators and in infinite Hamiltonian systems is to describe the completeness of eigenfunction systems.Traditional completeness is based on the spectral theory of self-adjoint operators.However,the infinite dimensional Hamiltonian operator is not self adjoint in general.At the end of the 20th century,Zhong Wanxie,a member of Chinese Academy of Sciences,extends the traditional method of separating variables by introducing Hamiltonian systems into elasticity.Then one can solve the problems in elasticity based on the completeness of eigenfunction systems of Hamiltonian operators.It is worth noting that the completeness of eigenfunction systems of Hamiltonian operators is closely connected with the symmetry of point spectrum with respect to about the imaginary axis.However,in general,the point spectrum of the infinite dimensional Hamiltonian operator is not necessarily symmetric with respect to imaginary axis.In order to solve this problem,we have to consider the symplectic self-adjointness of infinite dimensional Hamiltonian operators.In this dissertation,we are using the spectral properties of infinite dimensional Hamiltonian operators to obtain the sufficient conditions for Hamiltonian operators to be symplectic self-adjoint,and the necessary and sufficient conditions are obtained for some special Hamiltonian operators.On the other hand,the symmetry of point spectrum with respect to imaginary axis also plays an important role in proving spectral inclusion of Hamiltonian operators.Due to spectral inclusion,we can use the numerical range and numerical radius to characterize the spectral distribution of bounded linear operators.However,for unbounded linear operators,the spectral inclusion relation holds if and only if 1-type residual spectrum is contained in the closure of numerical range.The boundedness and convexity of the numerical range also play the important roles in spectral inclusion,but the numerical range is not necessarily a closed set.In 2013,D.Wu et al pointed out in their monograph if T is a compact operator,then the numerical range is closed if and only if it contains the origin.But we do not know when the origin belongs to the numerical range for general operators.To this end,we study the problem that the origin belongs to the numerical range,and answer the open problem proposed by P.Psarrakos,et al in 2003.we note that the radius of numerical range is continuous with respect to operator,so we study the problems of rational approximation and reciprocal approximation in Orlicz spaces,and obtain the corresponding estimates.Over the years,with the development of the local spectral theory emerged some powerful tools such as an important concept-single valued extension property-which have enriched the study of spectral structure of operators.In fact,there are many operators with single value extension property,including normal operators,spectral operators,generalized spectral operators and so on.Later,with the help of the concept of localized single valued extension property the close relationship has been established between B-Fredholm operators and Drazin invertible operators and pseudo B-Fredholm operators and generalized Drazin invertible operators,respectively.In this dissertation,apart from the single valued extension property of operator matrices,we obtain the relationship of pseudo B-Weyl spectrum and generalized Drazin spectrum.We improve the question that was not completely solved by H.Zariouh and H.Zguitti in 2016.And construct a counter-example to show the gap of Remark A(ⅲ)in the paper "Index of B-Fredholm operators and generalization of a Weyl theorem".