Inner, outer, and generalized inverses in banach and hilbert spaces

Abstract

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 ...