Research On Spectral And Geometric Structures Of Operators With Algebraic Characteristics

Operator theory and operator algebra is one of important research fields of functional analysis.Spectral theory has extensive applications in matrix theory,function theory,differential equation,control theory,quantum physics and other fields,hence the spectral structures,algebraic structures and geometric struc-tures of operator classes with different algebraic characteristics have become a hot topic for scholars.The operator satisfying Weyl type theorem is one type of impor-tant and classical operators,which can better reflect the distribution characteristics and geometric features of various spectrum.On the other hand,orthogonal projec-tion is an important research object in differential geometry,and then the operators related to orthogonal projections or constructed by orthogonal projections attract many scholars.In this paper,we consider the spectral structures and geometric structures of operators with some algebraic features.We mainly study the Weyl's theorem and(?)property for compact perturbation of 2 x 2 operator matrices,the pencil of orthogonal projections and von Neumann structure,also investigate the differential manifold structure and geodesics of the space of generalized projections.In Chapter 2,by the Fredholm index theory,we study the single value extension property,(?)property for all compact perturbations of 2 x 2 anti-diagonal operator matrix.According to the spectrum characteristics of all compact perturbations of operator,Weyl theorem for all compact perturbations of 2 x 2 upper triangular operator matrix is discussed.In Chapter 3,based on the spectral analysis of the pencil of orthogonal pro-jections,we give a matrix representation for self-adjoint operator T to be the pen-cil ?P + Q of a pair(P,Q)of orthogonal projections at some fixed real number? ? R\{-1,0},and then represent all pairs(P,Q)of orthogonal projections such that T = ?P + Q.Afterwards,the von Neumann algebra generated by such pairs(P,Q)and its commutant are characterized,and the pencil index of T is discussed.On the basis,we study the differential structure of the set of products of pair(P,Q)of orthogonal projections.In Chapter 4,we also concern the geometric structures of the set gp of gener-alized projections.Firstly,we study the differential manifold structure of gp with the aid of the unitary orbit under Banach-Lie group UA.Secondly,the existence and concrete form of geodesics of the given connection are focused.On this ba-sis,we concern the geodesic problem joining two given endpoints in gp.Beyond these,the geometric structures of product of gp and Banach-Lie group UA are also investigated.