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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 C. W. Groetsch, J. Math. Anal. Appl. 49, 154-157(1975; Zbl 0295.47012)], of the Drazin inverse [see Y. Wei and S. Qiao, Appl. Math. Comput. 138, 77-89 (2003; Zbl 1034.65037)], and of the specific generalized inverse studied by 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 Numerical solutions to equations with linear operators
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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