On the binary relations defined using GD1 and 1GD inverses over infinite dimensional vector spaces

TL;DR

This paper investigates binary relations based on GD1 and 1GD generalized inverses over infinite dimensional vector spaces, providing characterizations, algorithms, and order properties.

Contribution

It extends the theory of GD1 and 1GD generalized inverses to infinite dimensional spaces and characterizes their associated binary relations as partial orders.

Findings

Characterization of GD1 and 1GD relations via AST decomposition.

Algorithms for computing these generalized inverses.

Proof that these relations form partial orders on finite potent endomorphisms.

Abstract

The purpose of this article is to study certain binary relations of endomorphisms over infinite dimensional vector spaces defined by GD1 and 1GD generalized inverses. In order to do so, these generalized inverses are studied over arbitrary vector spaces (namely, infinite dimensional ones) using finite potent endomorphisms. We characterize them in terms of the AST decomposition of a finite potent endomorphism and we obtain algorithms for their respective computation. This theory is then used to characterize the GD1 and 1GD binary relations for finite potent endomorphisms in terms of the AST decomposition and to prove that they define partial orders in the set of finite potent endomorphisms, thus, completing the theory of these generalized inverses for matrices.