Moore-Penrose Inverse of some Linear Maps on Infinite-Dimensional Vector Spaces

TL;DR

This paper characterizes when the Moore-Penrose inverse exists for linear maps on infinite-dimensional inner product spaces and provides a method for computing it, with applications to solving infinite linear systems.

Contribution

It extends the concept of Moore-Penrose inverse to certain infinite-dimensional linear maps and offers a computational approach for these inverses.

Findings

Characterization of linear maps with Moore-Penrose inverse in infinite dimensions

Method for computing the Moore-Penrose inverse in infinite-dimensional spaces

Application to least norm solutions of infinite linear systems

Abstract

The aim of this work is to characterize linear maps of inner pro\-duct infinite-dimensional vector spaces where the Moore-Penrose inverse exists. This MP inverse generalizes the well-known Moore-Penrose inverse of a matrix $A\in \text{Mat}_{n\times m} ({\mathbb C})$. Moreover, a method for the computation of the MP inverse of some endomorphisms on infinite-dimensional vector spaces is given. As an application, we study the least norm solution of an infinite linear system from the Moore-Penrose inverse offered.