This paper develops a comprehensive theory of generalized inverse operators on Banach spaces. The paper is partly expository and partly new results. 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 obtain 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 proximinal 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. Algebraic complements to subspaces and algebraic projectors pla y a central role in the algebraic theory of generalized inverses.

Topological complements and topological projectors figure out prominently whenever issues related to analysis are raised in the theory of generalized inverses. Similarly notions related to metric projectors, in particular orthogonal complements and orthogonal projectors both in Hilbert and Banach spaces enter the scene whenever questions related to geometry (e.g., extremal and proximinal properties of generalized inverses) are posed, In contrast to most earlier results on generalized inverses of arbitrary linear operators in normed and Hilbert spaces, where sufficient conditions are usually stated to develop a particular theory, we show both the necessity and sufficiency of the conditions imposed for developing the algebraic, topological, projectional, metric proximinal, and extremal properties of generalized inverses. Bounded inner and bounded outer inverses are treated in detail and a brief section is devoted to the operator theory of the Drazin inverse.

Throughout we assume that the operator is neither one-to- one nor onto, so that all projectors and their complements used in this paper are nonzero. Occasionally we cite a reference in the manuscript by its date of publication, rather than by the number under which it is listed, to emphasize chronological order of development.