Halmos双射影理论及其相关问题的研究

双射影理论是算子论与算子代数的热点分支.申请人与合作者已对Halmos双射影理论与BarrySimon双射影超对称方法进行了探讨,得到一些结论.本课题首先拟运用算子分块技巧,解算子方程,算子谱理论等,给双射影理论更精细刻画,使其成为数学问题研究的新思路与更精细的工具;然后把Barry Simon双射影酉扰动定理延拓到可逆算子,考虑开问题:Barry Simon双射影酉扰动定理能否延拓到Hilbert空间非自伴射影对以及Banach空间的一般射影对.其次,运用双射影理论探讨正交射影及幂等算子乘积值域的几何结构,Krein空间的几何结构以及两子空间的夹角问题.最后,运用双射影理论到量子信息与薛定谔算子谱的研究中,关注Stan Gudder在量子效应方面的近期研究成果;考虑David Damanik教授的问题:极限周期薛定谔算子是否总有纯点谱.通过本项目研究,以期对相关领域产生积极影响.

The theory of two projections is an active research field of operator theory and operator algebra. It involves many research branches and applications of mathematics. The aplicant and collaborators have discussed the theory of two projections of Halmos and Bary Simon's supersymmetry method on two projections. At the same time, we have drawn some preliminary conclusions on the theory of two projections. In this project, firstly, we will get a more precise characteristic on the theory of two projections using block-operator technique, solving equations of operators and spectral theory of operators, which makes it a new idea and a more sophisticated tool for the study of some mathematical problems. In addition, we will extend the theorem of unitaries permuting two orthogonal projections of Bary Simon to the general two reversible operators. Further, we want to resolve a interesting open question which is whether there are extension of theorem of unitaries permuting two orthogonal projections of Bary Simon to non-self-adjoint Hilbert space projections and to general pairs of projections on a Banach space. . Secondly, we will study the geometry structures of range involving products of two orthogonal projections, products of two idempotent operators，the geometry structures of Krein spaces and the angle between two subspaces using two projections. . Thirdly, We will pay attention to Stan Gudder's research results on quantum information and the work of David Damanik on spectral theory of Schrodinger operators.On the one hand, we study quantum effect using two projections. On the other hand,we will consider the problem given by professor David Damanik about the spectral type of a limit-periodic operator being always pure. We hope that the research of this project will not only enrich theory of operator theory and operator algebras but also have a positive impact to study operator theory and operator algebras.