Moore-Penrose inverse and partial orders on Hilbert space operators

In this article we explore several aspects concerning to the Moore-Penrose inverse of a bounded linear operator. On the one hand, we study monotonicity properties of the Moore-Penrose inverse with respect to the Löwner, star, minus, sharp and diamond orders. On the other hand, we analyze the validity of the reverse order law, $B^{†}{A}^{†}=(AB{)}^{†}$, under hypothesis of operator ranges and also under hypothesis of order operators. Finally, we study the operator $B^{†}{A}^{†}$ as different weighted inverses of $AB$.