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The representation and approximations of outer generalized inverses. (English) Zbl 1071.65075
Let $X$ and $Y$ be Banach spaces, and $A\in L(X,Y)$. An operator $G\in L(X,Y)$ is called an outer generalized inverse $(OGI)$ of $A$ if $GAG=G$. A unified representation theorem for the class of all $OGI$’s of an operator is presented. The theorem is a generalization for the corresponding representation of the Moore-Penrose inverse [see {\it C. W. Groetsch}, J. Math. Anal. Appl. 49, 154-157(1975; Zbl 0295.47012)], of the Drazin inverse [see {\it Y. Wei} and {\it S. Qiao}, Appl. Math. Comput. 138, 77-89 (2003; Zbl 1034.65037)], and of the specific generalized inverse studied by {\it Y. Wei} [Linear Algebra Appl. 280, 87-96 (1998; Zbl 0934.15003)]. The unified representation is used to develop several particular expressions and computational procedures for the set of $OGI$’s. Some illustrative numerical examples are given.

##### MSC:
 65J10 Equations with linear operators (numerical methods) 47A05 General theory of linear operators
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