| Spectral theory is an important part of functional analysis.It has numerous applications in many parts of mathematics and physics including matrix theory,function theory,complex analysis,differential and integral equations,control theory and quantum physics.Research of the spectral theory has been for a long time,and it is becoming increasingly matured.The scope of application is gradually expand-ing.As a natural generalization of linear operators,the study of the linear relation spectrum theory is inevitable.In recent years,R.Cross and T.Alvarez et al are committed to generalize the Fredholm theory of linear operators to linear rclations.It has greatly enriched the research of spectral theory.The innovation and enrichment in research methods and contents of Local spectral theory and topological uniform descent to classical Fredholm theory oflinear operators,inspired us to discuss local spectral theory and topological uniform descent of linear relations.In this paper,we mainly study the generalization of the local spectral theory and the topological uniform descent of linear operators to the linear relations.The main results are two aspects:One is for the single valued extension property of linear relations.We properly define the single valued extension property of linear relation;We discuss the sin-gle valued extension property of semi-Fredhohm relation by means of the dimension of kernel and range of the linear relation,and we generalized some results about the single valued extension property of semi-Fredholni operators given by Finch to semi-Fredholm linear relations;We also give the characterizations of single val-ued extension property of the quasi-Fredholm relation by means of analytic core、quasi-nilpoternt part、ascent and descent of linear relaion,and we generalized the characterizations of single valued extension property at a point of quasi-Fredholim operators given by Aiena to quasi-Fredholm linear relations.On the other hand is to discuss about the topological uniform descent of linear relations.We properly define the topological uniform descent of linear relations;We generalized some results about the topological uniform descent of linear operators given by Grabiner to linear relations,and we get some characterization and structure theorems of topological uniform descent for linear relations. |
