On perturbations for oblique projection generalized inverses of closed linear operators in Banach spaces

The main concern of this paper is the perturbation problem for oblique projection generalized inverses of closed linear operators in Banach spaces. We provide a new stability characterization of oblique projection generalized inverses of closed linear operators under T-bounded perturbations, which improves some well known results in the case of the closed linear operators under the bounded perturbation or that the perturbation does not change the null space.     

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.     

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.     

Representations for Moore-Penrose inverses in Hilbert spaces

Let H₁, H₂ be two Hilbert spaces, and let T : H₁ → H₂ be a bounded linear operator with closed range. We present some representations of the perturbation for the Moore-Penrose inverse in Hilbert spaces for the case that the perturbation does not change the range or the null space of the operator.     

线性空间中代数广义逆的最简表示

本文主要在一般线性空间框架中从纯代数的角度研究代数广义逆的可加性与表示问题.首先在线性空间中利用空间代数直和分解给出I+AT~+可逆的充要条件,进而T~+=T~+(I+A~T+)~(-1),给出了T~+具有最简表示的一系列充要条件.其次讨论了在Banach空间广义逆和Hilbert空间Moore-Penrose逆扰动问题研究中的应用.本文的主要结果推广和改进了相关文献中的一些近期成果.     

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.     

Approximators to generalized inverses as regularizers of ill-posed problems

Let H₁ and H₂ be Hilbert spaces and let B be a closed linear operator mapping a dense subset of H₁ into H₂. Several families of approximators which converge t o the orthogonal or Moore-Penrose generalized inverse B^? are constructed. Additionally, the approximators are shown t o provide regularization operators for the equation Bx=y. Some of the results are extended to dissipative operators on reflexive and general Banach spaces.     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.     

On the semi-continuity of generalized inverses in Banach algebras

The main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a Banach algebra with identity. We first obtain a sufficient and necessary condition for the lower semi-continuity of reflexive generalized inverses as set-valued mappings. Based on this result, we characterize the continuity of the Moore-Penrose inverse in a C^∗-algebra and therefore, derive some new and well-known criteria in operator theory.     

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.     

On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements

The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and C^∗-algebra elements.     

Generalized inverses of circulant and generalized circulant matrices

An expression for the Moore-Penrose inverse of certain singular circulants by S.R. Searle is generalized to include all circulants. Similar expressions are given for the Moore-Penrose inverse of block circulants with circulant blocks, level-q circulants, k-circulants where |k|=1, and certain other matrices which are the product of a permutation matrix and a circulant. Expressions for other generalized inverses are given.     

Perturbation analysis for Moore-Penrose inverse of closed operators on Hilbert spaces

In this paper, we investigate the perturbation for the Moore-Penrose inverse of closed operators on Hilbert spaces. By virtue of a new inner product defined on H, we give the expression of the Moore-Penrose inverse $\bar{T}^{\dag}$ and the upper bounds of $\|\bar{T}^{\dag}\|$ and $\|\bar{T}^{\dag}-T^{\dag}\|$ . These results obtained in this paper extend and improve many related results in this area.     

Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities

In 1956, R. Penrose studied best-approximate solutions of the matrix equation AX = B. He proved that A⁺B (where A⁺ is the Moore-Penrose inverse) is the unique matrix of minimal Frobenius norm among all matrices which minimize the Frobenius norm of AX ? B. In particular, A⁺ is the unique best-approximate solution of AX = I. The vector version of Penrose's result (that is, the fact that the vector A⁺b is the best-approximate solution in the Euclidean norm of the vector equation Ax = b) has long been generalized to infinite dimensional Hilbert spaces.In this paper, an infinite dimensional version of Penrose's full result is given. We show that a straightforward generalization is not possible and provide new extremal characterizations (in terms of the Hermitian order) of A⁺ and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least-squares solutions. For a certain class of operators, we can phrase our characterizations in terms of a whole class of norms (including the Hilbert-Schmidt and the trace norms), thus providing new extremal characterizations even in the matrix case. We treat both operators with closed range and with not necessarily closed range. Finally, we characterize A⁺ as the unique inner inverse of minimal Hilbert-Schmidt norm if ∥A⁺∥₂ < ∞. We give an application of the new extremal characterization to the compensation problem in systems analysis in infinite-dimensional Hilbert spaces.     

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.     

Strong regularity and generalized inverses in jordan systems

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a?J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a?P(a)²J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a?J with generalized inverse b the subtriple generated by {a, b} is strongly associative     

On the characterization of generalized inverses by bordered matrices

A method to characterize the class of all generalized inverses of any given matrix A is considered. Given a matrix A and a nonsingular bordered matrix T of A,

$\text{T=}\text{A}\text{P}\text{Q}\text{R}$

the submatrix, corresponding to A, of T^-1 is a generalized inverse of A, and conversely, any generalized inverse of A is obtainable by this method. There are different definitions of a generalized inverse, and the arguments are developed with the least restrictive definition. The characterization of the Moore-Penrose inverse, the most restrictive definition, is also considered.     

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.     

Nonnegative Moore-Penrose inverses of operators over Hilbert spaces

The objective of this paper is to study the nonnegativity of the Moore-Penrose inverse of an operator between real Hilbert spaces. A sufficient condition ensuring this is given in terms of certain spectral property of all positive splittings of the given operator. A partial converse is proved.      

On generalized inverses and Green’s relations

We study generalized inverses on semigroups by means of Green’s relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse and Moore-Penrose inverse) belong to this class.