Banach空间中线性算子广义逆的连续性及其应用

研究了Banach空间中广义逆的扰动问题．给出了广义逆稳定的一些新特征,进而证明了这些稳定性特征与广义逆的选取无关,并由此得到了广义逆作为集值映射是下半连续的充要条件．     

Banach空间中线性算子的齐性广义逆

本文首先在Banach空间内引进拟线性投影算子的概念,由此给出Banach空 间内线性算子的齐性广义逆的统一定义。齐性广义逆包含线性广义逆、单值度量广义 逆.本文证得齐性广义逆存在的充分必要条件.     

正交投影列的强收敛准则与广义逆的Galerkin逼近

对Hilbert空间上的正交投影算子列,给出其强收敛的判别准则．设{Pn}是定义在某个Hilbert空间上的一列正交投影,此准则可描述为:{Pn}是强收敛的,当且仅当{R(Pn)}满足某种适当的条件(见文中的定理2．1),这里{R(Pn)}是与{Pn}相应的值域列．作为上述准则的应用,对有界线性算子,研究其正交广义逆的Galerkin逼近方法;这一研究给出了两条一般性结果(见文中的定理3．1和定理3．2),它们可应用于第一类算子方程的近似求解问题中．     

Approximate Methods for Generalized Inverses of Operators in Banach Spaces

In this paper, we present the necessary and sufficient condition of convergence of several iterative methods for computing the generalized inverses of operators in Banach spaces. It is proved that the iterative methods converge to the generalized inverse of an Operator in Banach spaces if and only if these conditions are satisfied.     

分块矩阵[A B]中一块的广义逆

给出矩阵[A B]的广义逆,其中A∈Cm×k,B∈Cm×(n-k),本文得到子块A的相关广义逆的计算公式.     

Hibert空间中广义框架的非交性

引入了Hilbert空间H中广义框架的非交性、强非交性,讨论了它们的一些性质;并且引入了保非交算子、强保非交算子,证明了酉算子、可逆算子是强保非交算子,下有界算子、余等距算子是保非交算子.     

Hilbert空间上线性算子的Drazin可逆性

主要研究了Hilbert空间上两个Drazin可逆算子和的Drazin可逆性.同时,对上三角算子矩阵的Drazin可逆性也给出了详细的讨论.     

Hilbert空间上线性算子广义逆A_(T,S)~((2))的存在性及其表示式

设H1和H2是两个Hilbert空间,B(H1,H2)表示从H1到H2的所有有界线性算子的集合,T和S分别是H1和H2的两个闭子空间．如果存在线性算子X∈B(H2,H1)满足XAX=X,R(X)=T,N(X)=S,则称X为线性算子A的具有指定像空间T和零空间S的外逆,记为AT,S(2)．该文进一步研究了线性算子广义逆AT,S(2)存在的若干等价条件及其性质,建立了算子广义逆AT,S(2)的表示形式．     

Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

Let H be a real Hilbert space and let T: H→2^H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit vT⁻¹0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.     

Generalized Polar Decompositions for Closed Operators in Hilbert Spaces and Some Applications

We study generalized polar decompositions of densely defined closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators. Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 and FRG-0456306, and the Austrian Science Fund (FWF) under Grant No. Y330.     

一些算子的交换子在广义Morrey空间上的紧性

本文主要研究分数次积分算子、Marcinkiewicz积分、带光滑核的pseudo-differential算子的交换子在广义Morrey空间M_p,ω（Rⁿ）上的紧性.注意它们的处理方法分别不同.     

赋范线性空间中渐近拟伪压缩型映象不动点的修改的广义Ishikawa迭代逼近

本文在去掉lim inf n→∞||xn||＜∞,∞∑n=0(kn-1)＜∞条件下,并用αn→o(n→∞)取代∑α2n＜∞,使用新的分析技巧,在赋范线性空间中建立了一致Lipschitz的渐近拟伪压缩型映象公共不动点的修改的广义Ishikawa迭代序列的强收敛定理,从而本质改进和推广了唐玉超,刘理蔚新近的结果.     

Hilbert空间中的广义松弛上强制变分不等式组

主要利用三步投影方法模型讨论了带误差估计的广义非线性上强制变分不等式组的逼近解及其收敛性,所得到结果推广和改进了一系列最新结果.     

Hilbert空间中的广义非线性变分不等式组的投影算法

利用投影技术讨论了Hilbert空间中一类含松弛伪上强制映射的广义非线性变分不等式组的逼近解及其收敛性,所得到结果推广和统一了系列最新结果.     

Banach空间中闭线性算子广义预解式存在定理

在Banach空间中研究闭线性算子广义逆扰动问题和广义预解式存在性问题.给出了闭线性算子广义逆在T-有界扰动下的一些稳定特征,这些特征推广了在有界线性算子情形、闭线性算子有界扰动情形以及闭线性算子保值域或保核空间情形的一些已知结果.以此为基础,得到了闭线性算子广义预解式存在的一些充要条件及其广义预解式的显式表达式.作为应用,给出了闭Fredholm算子和闭半-Fredholm算子的广义预解式存在性特征.     

Convergence Domains for Some Iterative Processes in Banach Spaces Using Outer or Generalized Inverses

We provide a semilocal Ptak–Kantorovich-type analysis for inexact Newton-like methods using outer and generalized inverses to approximate a locally unique solution of an equation in a Banach space containing a nondifferentiable term. We use Banach-type lemmas and perturbation bounds for outer as well as generalized inverses to achieve our goal. In particular we determine a domain such that starting from any point of our method converges to a solution of the equation. Our results can be used to solve undetermined systems, nonlinear least-squares problems, and ill-posed nonlinear operator equations in Banach spaces. Finally, we provide two examples to show that our results compare favorably with earlier ones.     

Finite-Dimensional Approximations of Operators in the Hilbert Spaces of Functions on Locally Compact Abelian Groups

A new approach to the approximation of operators in the Hilbert space of functions on a locally compact Abelian (LCA) group is developed. This approach is based on sampling the symbols of such operators. To choose the points for sampling, we use the approximations of LCA groups by finite groups, which were introduced and investigated by Gordon. In the case of the group R ⁿ , the constructed approximations include the finite-dimensional approximations of the coordinate and linear momentum operators, suggested by Schwinger. The finite-dimensional approximations of the Schrödinger operator based on Schwinger's approximations were considered by Digernes, Varadarajan, and Varadhan in Rev. Math. Phys. 6 (4) (1994), 621–648 where the convergence of eigenvectors and eigenvalues of the approximating operators to those of the Schrödinger operator was proved in the case of a positive potential increasing at infinity. Here this result is extended to the case of Schrödinger-type operators in the Hilbert space of functions on LCA groups. We consider the approximations of p-adic Schrödinger operators as an example. For the investigation of the constructed approximations, the methods of nonstandard analysis are used.     

Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

In this article, we prove optimal convergence rates results for regularization methods for solving linear ill-posed operator equations in Hilbert spaces. The results generalizes existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.     

Hilbert空间中强半压缩算子迭代序列的误差估计及稳定性分析

在实Hilbert空间中,本文讨论强半压缩算子下的迭代序列的误差估计式和强收敛定理,并举例与已有迭代序列的误差估计式进行对比.进一步地,分析该迭代序列的稳定性.我们的结果推广了许多已知结论.