Developing Reverse Order Law for the Moore–Penrose Inverse with the Product of Three Linear Operators

Abstract

In this paper, we study the reverse order law for the Moore–Penrose inverse of the product of three bounded linear operators in Hilbert spaces. We first present some equivalent conditions for the existence of the reverse order law ${\left(ABC\right)}^{†}=C^{†}{B}^{†}{A}^{†}$ . Moreover, several equivalent statements of $\mathrm{ℛ}\left(A{A}^{\ast }\left(ABC\right)\right)=\mathrm{ℛ}\left(ABC\right)$ and $\mathrm{ℛ}\left(C^{\ast }C{\left(ABC\right)}^{\ast }\right)=\mathrm{ℛ}\left({\left(ABC\right)}^{\ast }\right)$ are also deducted by the theory of operators.