度量广义逆的定性理论及广义逆在谱理论和Banach流形中的应用

The theory of generalized inverse has its genetic roots essentially in the context of so called “ill-posed”linear problems,there are some excellent applications in some fields of numerical linear algebra,optimization and control,statistics,and other areas of applied mathematics.Linear generalized inverses are not suitable for the study of the best approximately solutions of some ill-posed equations,then metric generalized inverses are investigated.Metric generalized inverses are nonlinear and set-valued generalized inverses.Using geometry theory of Banach spaces and the characteristic of stabilities of generalized inverses of operators,this project describes the existence,uniqueness and continuity of metric generalized inverses and applications in spectrum and Banach manifolds of generalized inverses.Firstly,existence,uniqueness of metric generalized inverses are investigated by approximately compact of geometrical properties for Banach spaces.Secondly,continuity and perturbations of metric generalized inverses are investigated by means of some theories of Banach Lemma for some operators of bounded homogeneous generalized orthogonal decomposition theorem etc.Moreover, using the theory of perturbations for generalized inverses and local fine points,square iand polynomial nvariance of narrow spectrum are investagated, on the other hand,the transversality theory with parameter for the mapping of Fredholm is generalized,after that,a kind of generalized transversality with parameter in Banach manifolds is given.

广义逆理论是在不适定的线性问题的背景下产生的，此理论在数值线性代数、优化与控制、统计学及应用数学等领域有重要应用. 由于线性广义逆对不适定方程最佳逼近解的研究不适用,因此展开了度量广义逆理论的研究.度量广义逆是一种非线性集值广义逆. 本项目以Banach空间中度量广义逆的存在性、唯一性、连续性,广义逆在谱理论和Banach流形中的应用为研究内容,以广义逆的稳定性特征与Banach空间几何理论为研究工具,1）利用逼近紧等Banach空间几何性质研究度量投影、度量广义逆的存在性、唯一性;2)运用有界齐性算子的Banach引理、广义正交分解定理等研究度量广义逆的连续性与扰动;3）应用广义逆的稳定性与局部精细点等理论,一方面给出狭义谱的平方不变性、多项式不变性，另一方面将Fredholm映射的带参数的横截性定理进行推广,给出一类Banach流形带参数的广义横截性定理.

本项目以广义逆的扰动理论等为研究内容,以广义逆的稳定性特征与Banach空间几何理论等为研究工具，1）在不假定算子的零空间上的度量投影的线性性质的情况下,给出了Banach 空间中线性算子的度量广义逆扰动分析，改进了之前的结果；2）应用广义Neumann引理，给出了Bannach空间上有界线性算子的Moore-Penrose度量广义逆的扰动分析；3）应用广义逆的稳定性与局部精细点等理论给出了非正则控制及在控制下寻求临界点的法则；4）总结并给出了一些能够用来帮助研究广义逆理论的Banach空间几何性质。