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Inner, outer, and generalized inverses in Banach and Hilbert spaces. (English) Zbl 0633.47001

From the author’s abstract: This paper develops a comprehensive theory of generalized inverse operators on Banach spaces. The expository part develops a unified theory of generalized inverses of linear (but not necessarily bounded) operators on normed spaces together with the additional properties that are obtained in Hilbert spaces. This part provides a simplification and an extension of the unified approach developed by Nashed and Votruba for several generalized inverses of linear operators on topological vector spaces. The new results deal with bounded inner and bounded outer inverses, new extremal and proximal properties and a few related observations and properties in several sections. The approach of this paper is to develop the theory of generalized inverses in Banach spaces starting from the well known algebraic theory of generalized inverses of an arbitrary linear transformation acting between vector spaces.
Contents: 1. Introduction. 2. Notation. 3. Least-squares solutions of linear operator equations. 4. Remarks on the Moore-Penrose inverse. 5. Orthogonal generalized inverses in Hilbert spaces. 6. New extremal characterizations of generalized inverses in Hilbert spaces. 7. Inner inverses and bounded inner inverses. 8. Outer inverses and bounded outer inverses. 9. Algebraic generalized inverses. 10. Generalized inverses in Banach spaces. 11. Best approximation and proximal properties of generalized inverses in Banach space. 12. The Drazin inverse of operators on Banach space. References.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A62 Equations involving linear operators, with operator unknowns
15A09 Theory of matrix inversion and generalized inverses
41A35 Approximation by operators (in particular, by integral operators)
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