Reverse order law for the Moore-Penrose inverse of closed-range adjointable operators on Hilbert \(C^\ast\)-modules (Q2829892)

Suppose that \(A\) and \(B\) are bounded adjointable operators between Hilbert C*-modules admitting bounded Moore-Penrose inverse operators. The author gives some necessary and sufficient conditions for the reverse order law \((AB)^{\dag} = B^{\dag}A^{\dag}\) to hold. If \(A\), \(B\) and \(AB\) have closed ranges, then the author shows that the following conditions are equivalent: {\parindent=0.6cm \begin{itemize} \item[(a)] \((AB)^{\dag} = B^{\dag}A^{\dag}\), \item [(b)] \( [ A^{\dag} A, BB^* ]=0\) and \([A^*A, BB^{ \dag}]=0\), \item [(c)] \(\mathcal{R}(A^*AB) \subset \mathcal{R}(B) \) and \(\mathcal{R}(BB^*A^*) \subset \mathcal{R}(A^*) \), and \item [(d)] \(A^*ABB^*\) is EP. NEWLINENEWLINE\end{itemize}} The part \((a) \Leftrightarrow (c)\) has already been proved by \textit{K. Sharifi} and \textit{B. A. Bonakdar} [Bull. Iran. Math. Soc. 42, No. 1, 53--60 (2016; Zbl 1373.46054)].