新型算子广义逆的理论分析及其在不适定算子方程中的应用

算子广义逆在数值分析、正则化和不适定问题中有重要的作用.本项目借助HilbertC*-模、Banach空间几何学和C*-代数等方面的基本理论,以及微分几何等方面的知识,研究(1)Banach空间新型算子广义逆(核逆、Core-EP逆和DMP逆等)的理论分析,包括特征表示式、微分性质和扰动分析.(2)新型算子广义逆的有效条件数研究.(3)研究Banach空间中算子Drazin逆的锐角(acute)扰动的等价条件和Hilbert空间中的算子Moore-Penrose广义逆的乘法扰动.(4)应用Hilbert空间中的算子新型广义逆(核逆、DMP逆、Core-EP逆),研究非定常迭代来求解奇异算子方程的商收敛性.将我们的理论结果运用于Hilbert空间中算子加权最小二乘解的扰动上界估计；新型算子广义逆用于求解不适定算子方程.本课题既有重要的理论意义，又有较广泛的实际应用背景.

The generalized inverses of operators play an important role in the numerical analysis,regularizations and ill-posed problems.With the help of Hilbert C*-module, Banach space geometry and C*-algebra,as well as the basic theory of the differential geometry,we study the following four topics: (1)Theoretical analysis of some new type generalized inverses ( Core inverse,Core-EP inverse and DMP inverse ) in Banach spaces, including characteristic expressions, differential properties and perturbations. (2) Effective condition numbers for some new type generalized inverses. (3) The equivalent condition of acute perturbation for the Drazin inverse of operators in Banach space;Multiplication perturbation of the Moore-Penrose inverse of operators in Hilbert space.(4) Applications of some new type generalized inverses to the quotient convergence of singular operator equations, the upper bounds for the solutions of operator weighted least squares problem in the Hilbert space, as well as to solution of the ill-posed operator equations.This project is of important theoretical significances and wide practical backgrounds.