On the product of projectors and generalized inverses

TL;DR

This paper investigates conditions under which the product of projectors and generalized inverses remains a generalized inverse, providing new representations and focusing on applications to matrices and boundary problems.

Contribution

It introduces new criteria for the product of projectors to be a projector, using kernel and image properties, and offers a novel representation of generalized inverses without explicit factor knowledge.

Findings

Characterization of when the product of projectors is a projector

New representation of generalized inverses independent of explicit factors

Applications to matrices and boundary value problems

Abstract

We consider generalized inverses of linear operators on arbitrary vector spaces and study the question when their product in reverse order is again a generalized inverse. This problem is equivalent to the question when the product of two projectors is again a projector, and we discuss necessary and sufficient conditions in terms of their kernels and images alone. We give a new representation of the product of generalized inverses that does not require explicit knowledge of the factors. Our approach is based on implicit representations of subspaces via their orthogonals in the dual space. For Fredholm operators, the corresponding computations reduce to finite-dimensional problems. We illustrate our results with examples for matrices and linear ordinary boundary problems.