Developing reverse order law for the Moore-Penrose inverse with the product of three linear operators (Q2051685)

Summary: 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 \((ABC)^\dagger= C^\dagger B^\dagger A^\dagger\). Moreover, several equivalent statements of \(\mathscr{R}(AA^\ast (ABC))=\mathscr{R}(ABC)\) and \(\mathscr{R}(C^\ast C(ABC)^\ast)=\mathscr{R}((ABC)^\ast)\) are also deducted by the theory of operators.