Study On Some Problems Of Spectral Perturbations For Linear Relations

With the further research of operator theory(for the theory of linear op-erators,readers are referred to the books[1,21,34,37,39,43,47,52,56,60,77,81,82]),more and more multi-valued operators and non-densely defined operators have been found.For example,the operators generated by those linear continuous Hamiltonian systems,which do not satisfy the defi-niteness conditions,and general linear discrete Hamiltonian systems may be multi-valued or not densely defined in their corresponding Hilbert spaces(cf.,[49,58,59,70]).So the classical theory of operators is not available in this case.Motivated by the need to consider the operators of this kind as well as enrichment of many aspects of operator theory,it is necessary and urgent to establish theory of multi-valued operators and non-densely defined Hermitian operators.Single-valued and multi-valued linear operators are called linear relations or linear subspaces,briefly,relations or subspaces.In this paper,we study several problems of spectral perturbations for linear relations,including the variation of spectra of closed linear relations under some perturbations in Banach spaces;the stability of essential spectra of self-adjoint relations under relatively compact perturbations;the stability of absolutely continuous spectra of self-adjoint relations under trace class perturbations.We first study the stability of spectra of closed linear relations under some perturbations in Banach spaces.In 1966,Kato[47]proved that the spectrum of a closed operator in a Banach space is upper semi-continuous,which means that the spectrum does not expand suddenly when the operator is continuously changed.However,it may shrink suddenly.Further,he gave out some error estimates of spectra of closed operators in Banach spaces and self-adjoint operators in Hilbert spaces under bounded perturbation,respectively.If these results hold for the multi-valued linear operators case?Motivated by Kato’s works,we introduce a concept of upper-continuity of the spectrum of a closed linear relations,and investigate the variation of spectra of closed relations under some perturbations.In the special case that the linear relation is a self-adjoint relation,we shall discuss the perturbations of its spectrum.We then study the stability of essential spectra of self-adjoint relations under relatively compact perturbations.Currently,there are several defini-tions for essential spectra of linear relations,and two of them are common.One is based on the semi-Fredholm properties of a linear relation to give dif-ferent definitions(cf.,[28,83]),and the other is defined as the subset of its spectrum consisting of either accumulation points or isolated eigenvalues of infinite multiplicity[69].The perturbation theory of essential spectra of linear relations has been investigated.In 1998,Cross[28]introduced concepts of relative boundedness and relative compactness of linear relations,and showed that one kind of essential spectrum of a linear relation is stable under rela-tively compact perturbation with certain additional conditions.In 2014,Wicox[83]gave five distinct essential spectra of linear relations in Banach spaces in terms of semi-Fredholm properties,and showed their stability under relatively compact perturbation with some additional conditions and under compact per-turbation,separately.In 2016,Shi[66]showed that under relatively compact perturbation with some additional conditions,the essential spectrum of a self-adjoint linear relation defined by the second definition is invariance.Motivated by Weidmann’s methods[81],we discuss the stability of the essential spectrum of a self-adjoint relation defined by the second definition under relatively compact or more general perturbations.To the best of our knowledge,there seem a few results about perturbations of absolutely continuous spectra of self-adjoint relations.However,the perturbation theory of absolutely continuous spectra of self-adjoint operators has been extensively studied and some elegant results have been obtained(cf.,[47,81]).A classical result of them is that the absolutely continuous spectrum of a self-adjoint operator is stable under trace class perturbations[47].Can we extend this result to self-adjoint relations?This is the last problem which we shall discussed in this paper.In 1961,Arens[11]showed that every closed linear relation T in a Hilbert space can be decomposed as an operator part T_s and a purely multi-valued part T∞.In 1985,Dijksma with his coauthor[32]proved that if a linear relation T in a Hilbert space is self-adjoint,then its operator part T_s is also self-adjoint in the Hilbert spaces T(0)~⊥.Later,Shi studied some spectral properties of self-adjoint relations together with her coauthors,and showed that the absolutely continuous spectra of a self-adjoint relation and its operate part are identical[69].Using these results,we introduce a concept of trace class relations,and discuss the relationships between the perturbations of a closed relation and the corresponding perturbations of its operator part,and the relationships between the spectra of its perturbation and the spectra of its operator part’s corresponding perturbation,respectively.Then,we discuss the stability of absolutely continuous spectra of self-adjoint relations under trace class perturbations.The following is the organization of this paper.This dissertation is divided into five chapters.In Chapter 1,some ba-sic.concepts and fundamental results about linear relations are introduced,and the spectra and the classifications of the spectra of linear relations are recalled.Further,we give some perturbations of linear relations,including small gap perturbation,relatively bounded perturbation,relatively compact perturbation,degenerate and trace class perturbations.In Chapter 2,we pay attention to the variation of spectra of closed linear relations in Banach spaces under some perturbations.Firstly,we shall introduce a concept of commutativity for linear relations.Then,it is shown that the boundedness of the inverse of a closed linear relation is preserved under relatively bounded and small gap perturbations,and perturbation about spectral condition,separately.By using these results,upper semi-continuous of spectra of closed linear relations in obtained and an error estimate of spectra of closed relations is given under bounded perturbation.In addition,the stability of self-adjointness of a self-adjoint linear relation under relatively bounded and small gap perturbations,and the variation of spectrum of a self-adjoint linear relation under bounded perturbation are discussed.Chapter 3 is devoted to stability of essential spectra of self-adjoint rela-tions under relatively compact perturbation in Hilbert spaces.Firstly,some relationships between relative boundedness and relative compactness of linear relations are established,and some sufficient and necessary conditions of them are given.Then,it is shown that the essential spectra of self-adjoint relations are invariant under either relatively compact perturbation or a more general perturbation.These results generalize the corresponding results of self-adjoint operators,and some of which weaken certain assumptions of the related exist-ing results.In Chapter 4,we study stability of absolutely spectra of self-adjoint relations under trace class perturbation in Hilbert spaces.Firstly,we introduce a new linear operator A_T induced by two linear relations T and A,which plays an important role in this chapter,and then study its some properties.In the case that T is closed,we investigate the relationships between the relation T + A and the operator T_s + A_T,including their closedness,Hermiteness,self-adjointness and spectra.Then,we establish some relationships between the perturbations of T and the corresponding perturbations of T_s.Using these results,we investigate stability of absolutely spectra of self-adjoint relations under trace class perturbation.Chapter 5 is the conclusion,including the main results we have obtained,their meanings,and an outlook of our future study.