Spectral Properties Of Linear Relation Matrices

The linear relation was first introduced to consider adjoints of nondensely defined linear differential operators by American mathematician von Neumann J.,and it has important applications in nonlinear analysis,extension theory of linear operators,differ-ential equations,degenerate operator semigroups,and optimization and control problems.Recently,the spectral properties of linear relation matrices begin to attract authors,at-tention.This dissertation deals with the spectral properties of linear relation matrices in a Hilbert space by using the properties of multi-valued parts and selections of linear rela-tions and the space decomposition method.Particularly,the perturbations of spectra of relation matrices are discussed,and the essential spectrum,Weyl spectrum,essential ap-proximate point spectrum,Browder spectrum and Browder essential approximate point spectrum of upper triangular relation matrices are characterized by the corresponding spectra of its diagonal entries.First,for given linear relations A,B,denote by MC=(?)the upper triangular linear relation matrix.A necessary and sufficient condition is presented under which the range of MC is(not)closed for an arbitrary bounded linear operator C based on the finite rank perturbation of the range of relations and the space decomposition method.On this basis,the perturbation of the closed range spectrum for upper triangular relation matrices is obtained.Moreover,we also discuss the necessary and sufficient condition for MC to have the properties above when C is runs over bounded linear relations.Next,we use the relationship between the properties of MC and QMC MC to ob-tain the necessary and sufficient conditions for MC to be a Fredholm relation,left(right)Fredholm relation,Weyl relation and left(right)Weyl relation for some relation C,respec-tively.Then the perturbations of essential spectrum,left(right)essential spectrum,Weyl spectrum,essential approximate point spectrum(left Weyl spectrum)and right Weyl spectrum for upper triangular relation matrices are further given.In addition,for given linear relations A,B,C,we characterize ?*(MC)by the block entries of upper triangular relation matrix MC,and determine the set W such that ?*(A)? ?*(B)=?*(Mc)? W holds,where ?*? {?e,?w,?ea,?b,?ab},and the notations in the set denote the essential spectrum,Weyl spectrum,essential approximate point spectrum,Browder spectrum and Browder essential approximate point spectrum.The Fredholmness result is applied to the linear relation constructed by a Hamiltonian operator matrix derived from the plate bending equation.Then,the perturbations of the spectrum,point spectrum,residual spectrum and continuous spectrum for the upper triangular linear relation matrix MC are characterized based on the space decomposition method and the properties of selections of linear rela-tions.When A,B,C are all given,the point spectrum of MC is characterized by the properties of its entries.Additionally,the relationship between the spectrum of MC and the union of spectrum of diagonal elements is described under the local spectral theory.Finally,for the given linear relations A,B,C,write linear relation matrix MX=(?).A necessary and sufficient condition is given for relation matrix Mx to be right(left)invertible and invertible relations for some bounded linear operator X,respectively.At the same time,we also discuss the problems for MX to be left(right)invertible and invertible for some linear relation X,and the perturbations of left(right)spectrum and spectrum of MX are further obtained.