Algebraic properties of Moore-Penrose inverse operators

Abstract

This article elucidates the algebraic attributes of Moore-Penrose inverse operators on Hilbert spaces. We establish requisite conditions for normality and derive properties such as the reverse order law and invertibility. Unitary behavior is demonstrated under specific circumstances, while quasi-unitariness is explored, yielding novel insights. Illustrative examples using integer matrices and Bergman space operators are also provided to demonstrate the application and validity of the obtained results.