Moore-Penrose inverse and doubly commuting elements in $C^*$-algebras

TL;DR

This paper investigates the Moore-Penrose inverse in $C^*$-algebras, establishing conditions under which the reverse order law holds for doubly commuting elements and exploring various applications.

Contribution

It provides new characterizations of the reverse order law for Moore-Penrose inverses in $C^*$-algebras, especially for doubly commuting regular elements.

Findings

Reverse order law holds for the Moore-Penrose inverse of doubly commuting regular elements.

Conditions are identified for when the reverse order law characterizes doubly commuting elements.

Applications include normal $C^*$-algebra elements, Hilbert space operators, and Calkin algebras.

Abstract

In this work it is proved that the Moore-Penrose inverse of the product of $n$-doubly commuting regular $C^*$-algebra elements obeys the so-called reverse order law. Conversely, conditions regarding the reverse order law of the Moore-Penrose inverse are given in order to characterize when $n$-regular elements doubly commute. Furthermore, applications of the main results of this article to normal $C^*$-algebra elements, to Hilbert space operators and to Calkin algebras will be considered.