Perturbation analysis for oblique projection generalized inverses of closed linear operators in Banach spaces

In this paper, we study the perturbation problem for oblique projection generalized inverses of closed linear operators in Banach spaces. By the method of the perturbation analysis of linear operators, we obtain an explicit perturbation theorem and error estimates for the oblique projection generalized inverse of closed linear operators under the T-bounded perturbation, which extend the known results on the perturbation of the oblique projection generalized inverse of bounded linear operators in Banach spaces.     

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore–Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore–Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore–Penrose metric generalized inverse of bounded linear operators in Banach spaces.     

Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T⁺(I+δTT⁺)^-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T^†(I+δTT^†)^-1. The results obtained in this paper extend and improve many recent results in this area.     

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.     

Banach空间中基于Neumann引理的拟线性广义逆扰动定理

本文给出Banach空间中闭线性算子的Moore-Penrose有界拟线性投影广义逆的一种新的扰动分析方法.运用的核心工具是广义Neumann引理,这与以往其他结果中所运用的广义Banach引理的处理方法极为不同,得到了闭线性算子的MoorePenrose有界拟线性广义逆的又一个扰动定理及三个误差界不等式.     

Perturbations of Moore–Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper,the perturbations of the Moore–Penrose metric generalized inverses of linear operators in Banach spaces are described.The Moore–Penrose metric generalized inverse is homogeneous and nonlinear in general,and the proofs of our results are different from linear generalized inverses.By using the quasi-additivity of Moore–Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition,we show some error estimates of perturbations for the singlevalued Moore–Penrose metric generalized inverses of bounded linear operators.Furthermore,by means of the continuity of the metric projection operator and the quasi-additivity of Moore–Penrose metric generalized inverse,an expression for Moore–Penrose metric generalized inverse is given.     

Inner Generalized Inverses with Prescribed Idempotents

We define and characterize inner generalized inverses with prescribed idempotents. These classes of generalized inverses are natural algebraic extension of generalized inverses of linear operators with prescribed range and kernel. We consider the reverse order rule for inner generalized inverses of elements of a ring, some perturbation bounds, and we construct an iterative method for computing a (p, q)-inner inverse in Banach algebras.     

A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES

In this paper,we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is "the generalized Neumann lemma" which is quite different from the method in [12] where "the generalized Banach lemma" was used. By the method of the perturbation analysis of bounded linear operators,we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.     

On Some Applications of Geometry of Banach Spaces and Some New Results Related to the Fixed Point Theory in Orlicz Sequence Spaces

We present some applications of the geometry of Banach spaces in the approximation theory and in the theory of generalized inverses. We also give some new results, on Orlicz sequence spaces, related to the fixed point theory. After a short introduction, in Section 2 we consider the best approximation projection from a Banach space $X$ onto its non-empty subset and proximinality of the subspaces of order continuous elements in various classes of Köthe spaces. We present formulas for the distance to these subspaces of the elements from the outside of them. In Section 3 we recall some results and definitions concerning generalized inverses of operators (metric generalized inverses and Moore-Penrose generalized inverses). We also recall some results on the perturbation analysis of generalized inverses in Banach spaces. The last part of this section concerns generalized inverses of multivalued linear operators (their definitions and representations). The last section starts with a formula for modulus of nearly uniform smoothness of Orlicz sequence spaces $\ell^\Phi$equipped with the Amemiya-Orlicz norm. From this result a criterion for nearly uniform smoothness of these spaces is deduced. A formula for the Domínguez-Benavides coefficient $R(a,l_\Phi)$ is also presented, whence a sufficient condition for the weak fixed point property of the space $\ell^\Phi$is obtained.     

Mixed Boundary Value Problems for the Helmholtz Equation in a Quadrant

The main objective is the study of a class of boundary value problems in weak formulation where two boundary conditions are given on the half-lines bordering the first quadrant that contain impedance data and oblique derivatives. The associated operators are reduced by matricial coupling relations to certain boundary pseudodifferential operators which can be analyzed in detail. Results are: Fredholm criteria, explicit construction of generalized inverses in Bessel potential spaces, eventually after normalization, and regularity results. Particular interest is devoted to the impedance problem and to the oblique derivative problem, as well.     

Perturbation of the Drazin inverse for closed linear operators

We investigate the perturbation of the Drazin inverse of a closed linear operator recently introduced by second author and Tran, and derive explicit bounds for the perturbations under certain restrictions on the perturbing operators. We give applications to the solution of perturbed linear equations, to the asymptotic behaviour ofC ₀-semigroups of linear operators, and to perturbed differential equations. As a special case of our results we recover recent perturbation theorems of Wei and Wang.     

Analytic perturbation of generalized inverses

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. We provide an algorithm for computing the coefficients of the Laurent series of the perturbed Drazin generalized inverse. In particular, the regular part coefficients can be efficiently calculated by recursive formulae. Finally we apply the obtained results to the perturbation analysis of the Moore–Penrose generalized inverse in the real domain.     

Multivalued linear projections

A multivalued linear projection operator P defined on linear space X is a multivalued linear operator which is idempotent and has invariant domain. We show that a multivalued projection can be characterised in terms of a pair of subspaces and then establish that the class of multivalued linear projections is closed under taking adjoints and closures. We apply the characterisations of the adjoint and completion of a projection together with the closed graph and closed range theorems to give criteria for the continuity of a projection defined on a normed linear space. A new proof of the theorem on closed sums of closed subspaces in a Banach space (cf. Mennicken and Sagraloff [9, 10]) follows as a simple corollary. We then show that the topological decomposition of a space may be expressed in terms of multivalued projections. The paper is concluded with an application to multivalued semi-Fredholm relations with generalised inverses.     

Weighted Generalized Inverses,Oblique Projections,and Least-Squares Problems

A generalization with singular weights of Moore–Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.     

Weighted binary relations for operators on Banach spaces

Using the Wg-Drazin inverses, we introduce and characterize new weighted pre-orders on the set of all bounded linear operators between two Banach spaces. As an application, we present two generalized Drazin pre-orders and an extension of the generalized Drazin order to a partial order.     

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.     

Criteria for the Metric Generalized Inverse and its Selections in Banach Spaces

We investigate the metric generalized inverses of linear operators in Banach spaces and their homogeneous selections, which was the research suggestion given by Nashed and Votruba (Bull. Am. Math. Soc. 80:831–835, 1974). We construct a kind of the bounded homoneneous selections for the set-valued metric generalized inverse. Criteria for the metric generalized inverses of linear operators and their homogeneous selections are given in terms of Moore–Penrose conditions. The research was supported in part by the National Science Foundation Grant (10671049) and the Science Foundation Grant of Heilongjiang Province.     

The Algebraic Perturbation Method for Generalized Inverses

Algebraic perturbation methods were first proposed for the solution of nonsingular linear systems by R. E. Lynch and T. J. Aird [2]. Since then, the algebraic perturbation methods for generalized inverses have been discussed by many scholars [3]-[6]. In [4], a singular square matrix was perturbed algebraically to obtain a nonsingular matrix, resulting in the algebraic perturbation method for the Moore-Penrose generalized inverse. In [5], some results on the relations between nonsingular perturbations and generalized inverses of $m\times n$ matrices were obtained, which generalized the results in [4]. For the Drazin generalized inverse, the author has derived an algebraic perturbation method in [6]. In this paper, we will discuss the algebraic perturbation method for generalized inverses with prescribed range and null space, which generalizes the results in [5] and [6]. We remark that the algebraic perturbation methods for generalized inverses are quite useful. The applications can be found in [5] and [8]. In this paper, we use the same terms and notations as in [1].     

Operator part of infinite system of differential equations

Let T be a differential operator in a Hilbert space generated by a first-order infinite system of an ordinary linear differential expression, which is subject to infinitely many boundary conditions. We solve completely for y from Ty = g. The main tools involved are operator parts of closed linear manifolds, which are closely related to generalized inverses.     

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.