Reverse Order Law for Closed Range Operators in Hilbert Spaces

TL;DR

This paper investigates the conditions under which the reverse order law holds for Moore-Penrose inverses of closed range operators in Hilbert spaces, providing over 50 theoretical results and examples of failure cases.

Contribution

It offers a comprehensive set of new results characterizing the reverse order law for Moore-Penrose inverses in infinite-dimensional Hilbert spaces, including range inclusion conditions.

Findings

Over 50 new theoretical results established

Range inclusion conditions for reverse order law identified

Examples illustrating failure cases in infinite dimensions

Abstract

We present more than 50 results including some range inclusion results to characterize reverse order law for Moore-Penrose inverse of closed range Hilbert space operators. We use basic properties of Moore-Penrose inverse to prove the results. Some examples are also provided to illustrate failure cases to hold the reverse order law in infinite dimensional settings.