Research On Operator Spectral Structure And Orthogonal Projection Pairs

Operator theory is one of the important research fields of functional analysis,It has numerous applications in many parts of differential equations,harmonic analysis and the-oretical physics.Spectral perturbations,preserver problems and pairs of orthogonal pro-jections are the hot topics in operator theory.As we know,Weyl’s theorem of the operator can reflect the characteristics of the spectrum of the operator,so Weyl’s theorem and it-s variation have caused the extensive concern.Also,many authors are interested in the study of linear or additive maps preserving the spectrum as well as certain parts of spec-trum.Moreover,based on Halmos’ two projections theorem,many authors have studied pairs of orthogonal projections with spectral theory and Fredholm theory,and got some results about pairs of orthogonal projections with a fixed difference.These results have great influence on operator theory,there are still some problems to be solved.In this paper,we discuss Weyl’s theorem and its variation,additive surjective maps preserving certain parts of spectrum and pairs of orthogonal projections with a fixed difference.Our main results can be divide into three parts.In the study of spectral perturbations,using the characteristics of the semi-Fredholm domain,we study the stability of Weyl’s theorem,property(ω).Moreover,the single valued extension property for 2 x 2 upper triangular operator matrices are considered.In the preserver problems,we firstly define m-normal eigenvalue,then discuss m-normal eigenvalue and m + 1-normal eigenvalue can be regarded as invariants of an au-tomorphism or an anti-automorphism on the algebra of all bounded linear operators.In addition,we characterize that an additive surjective map φ on B(X)preserving the semi-Fredholm domain in spectrum.Moreover,we study the linear maps preserving topolog-ical uniform descent and preserving the stability of the single valued extension property respectively.Moreover,we investigate the self-adjoint operators which are a difference of two orthogonal projections.Firstly,we study the case that A is in the generic position,we give a sufficient and necessary condition for A to be a difference of a pair of orthogonal projections.We give a characterization of all pairs(P,Q)of orthogonal projections such that A = P-Q.Then we extend the results to the general case.Finally,we characterize the von Neumann algebra generated by such pairs(P,Q).