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
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