Moore-Penrose inverse and doubly commuting elements in $C^{\ast }$-algebras

Abstract

In this work it is proved that the Moore-Penrose inverse of the product of $n$-doubly commuting regular $C^{\ast }$-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 $C^{\ast }$-algebra elements, to Hilbert space operators and to Calkin algebras will be considered.

arXiv comment12 pages, original research article