拟线性算子广义逆的扰动及其在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.Using geometry theory of Banach spaces and the characteristic of stabilities of generalized inverses of operators,this project describes the perturbations and stabilities of quasi-linear generalized inverses of bounded linear operators and generalized transversality with parameter in Banach manifolds.Firstly,Banach lemma of linear operators will be generalized to bounded homogeneous operators.Perturbations and stabilities of quasi-linear generalized inverses of bounded linear operators are investigated by means of some theories of generalized orthogonal decomposition theorem and the property of quasi-additive of quasi-linear projector etc.Moreover,expressed in the form to quasi-linear generalized inverses and some estimates of the norm and error are given.Secondly,a rank theorem of Fredhoml operators between Banach manifolds is introduced.We generalized the transversality theorem with parameter by means of a characteristic of stability of generalized inverse.Furthermore,a kind of generalized transversality with parameter in Banach manifolds is given.

算子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. 本项目以Banach空间中拟线性算子广义逆的扰动及稳定性与Banach流形带参数的广义横截性为研究内容，以Banach空间几何理论与算子广义逆的稳定性特征为研究工具，(1)将线性算子的Banach引理推广到有界齐性算子，应用Banach空间中广义正交分解定理、拟线性投影的拟可加性等理论研究拟线性广义逆的扰动及稳定性，进而给出拟线性广义逆的表示及范数估计和误差估计；(2) 给出Banach流形间Fredholm映射的秩定理，应用广义逆稳定性的特征将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.

广义逆理论是在不适定的线性问题的背景下产生的,是研究不适定方程解集度量结构、非线性方程解集的微分结构的特征及从非单特征值出发的分歧解的重要工具,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. 本项目以应用Banach空间几何理论与算子理论相结合研究算子广义逆理论为研究方向,主要以拟线性算子广义逆——Banach空间中有界线性算子的Moore-Penrose度量广义逆的扰动为研究内容.本项目在项目执行期间得到了Moore-Penrose 度量广义逆的扰动定理. 本项目将线性算子的Banach引理推广到齐性算子,应用Banach空间中广义正交分解定理、算子值域的逼近紧性、拟线性投影的拟可加性等理论,证明了Banach空间上有界线性算子的Moore-Penrose度量广义逆是具有连续性的拟线性的算子,给出了算子扰动后的Moore-Penrose度量广义逆的表达式及误差估计. 在线性算子广义逆扰动理论不适用于拟线性算子广义逆的研究、拟线性广义逆理论的研究还处于初始阶段的情况下,本项目应用 Banach空间几何性质、广义正交分解定理等理论,研究了Moore-Penrose度量广义逆的扰动.本项目的实施将有助于广义逆理论统一框架的形成,为拟线性广义逆的选择、稳定性提供了一个可行的途径,完善了非线性算子广义逆的稳定性理论,进一步,为接下来研究Banach流形的结构奠定了基础.