Abstract:
In this paper, we first study the perturbations and expressions for the generalized inverses $a^{(2)}_{p,q}$, $a^{(1, 2)}_{p,q}$, $a^{(2, l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate the general perturbation analysis and error estimate for some of these generalized inverses when $p,\,q$ and $a$ also have some small perturbations.

Abstract:
We investigate the stable perturbation of the generalized Drazin inverses of closed linear operators in Banach spaces and obtain some new characterizations for the generalized Drazin inverses to have prescribed range and null space. As special cases of our results, we recover the perturbation theorems of Wei and Wang, Castro and Koliha, Rakocevic and Wei, Castro and Koliha and Wei.

Abstract:
We study the natural inverse introduced by X. Mary and show some connections with the $(p,q)$-inverses of Djordjevic and Wei, where $p$ and $q$ are prescribed idempotents. We deal first with rings with identity and then specialize to the particular case of the algebra of bounded linear operators. We give a characterization of the set of operators along which an operator is natural invertible in terms of prescribed range and nullspace. Finally, the special case when the prescribed idempotent is the spectral projection is discussed.

Abstract:
Let and be Banach spaces, and let be a bounded linear operator. In this paper, we first define and characterize the quasi-linear operator (resp., out) generalized inverse (resp., ) for the operator , where and are homogeneous subsets. Then, we further investigate the perturbation problems of the generalized inverses and . The results obtained in this paper extend some well-known results for linear operator generalized inverses with prescribed range and kernel. 1. Introduction and Preliminaries Let and be Banach spaces, let be a mapping, and let be a subset of . Recall from [1, 2] that a subset in is called to be homogeneous if for any and , we have . If for any and , we have , then we call as a homogeneous operator on , where is the domain of ; is called a bounded homogeneous operator if maps every bounded set in into bounded set in . Denote by the set of all bounded homogeneous operators from to . Equipped with the usual linear operations for , and for , the norm is defined by , and then similar to the space of all bounded linear operators from to , we can easily prove that is a Banach space (cf. [2, 3]). Throughout this paper, we denote by , , and the domain, the null space, and the range of a bounded homogeneous operator , respectively. Obviously, we have . For an operator , let and be closed subspaces of and , respectively. Recall that the out inverse with prescribed range and kernel is the unique operator satisfying . It is well known that the important kinds of generalized inverses, the Moore-Penrose inverse, the Drazin inverse, the group inverse, and so on, are all generalized inverse (cf. [4, 5]). Researches on the generalized inverse of operators or matrices have been actively ongoing for many years (see [5–12], e. g.). Let and let and be two homogeneous subsets in and , respectively. Motivated by related work on in the literature mentioned above and by our own recent research papers [13, 14], in this paper, we will establish the definition of the quasi-linear operator outer generalized inverse with prescribed range and kernel . We give the necessary and sufficient conditions for the existence of the generalized inverses , and we will also study the perturbation problems of the generalized inverse . Similar results on the generalized inverse are also given. 2. Definitions and Some Characterizations of and We first give the concepts of quasi-additivity and quasi-linear projectors in Banach spaces, which are important for us to present the main results in this paper. Definition 1. Let be a subset of and let be a mapping. Ones calls as

Abstract:
Let $R$ be a ring and $b, c\in R$. In this paper, we give some characterizations of the $(b,c)$-inverse, in terms of the direct sum decomposition, the annihilator and the invertible elements. Moreover, elements with equal $(b,c)$-idempotents related to their $(b, c)$-inverses are characterized, and the reverse order rule for the $(b,c)$-inverse is considered.

Abstract:
The properties and some equivalent characterizations of equal projection(EP), normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R, then a is EP if and only if aaa#=a#aa. At the same time, the equivalent characterizations of a regular element to be EP are discussed. Let a∈R, then there exist b∈R such that a=aba and a is EP if and only if a∈R, a=ababa. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R, then a is normal if and only if a*a=aaa*. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R∩R#, n∈N, then a is normal if and only if a*a(a#)nn=a#a*(a)nn. The results generalize the conclusions of Mosi et al.

Abstract:
This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi-inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.

Abstract:
Several characterizations of EP and normal Moore-Penrose invertible Banach algebra elements will be considered. The Banach space operator case will be also studied. The results of the present article will extend well known facts obtained in the frames of matrices and Hilbert space operators.

Abstract:
Let $X, Y$ be Banach spaces and $T : X \to Y$ be a bounded linear operator. In this paper, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse $T^h$ and quasi--linear projector generalized inverse $T^H$ of $T$. Some applications to the representations and perturbations of the Moore--Penrose metric generalized inverse $T^M$ of $T$ are also given. The obtained results in this paper extend some well--known results for linear operator generalized inverses in this field.