On the Moore-Penrose inverse in $C^*$-algebras

In this article, two results regarding the Moore-Penrose inverse in the frame of $C^*$-algebras are considered. In first place, a characterization of the so-called reverse order law is given, which provides a solution of a problem posed by M. Mbekhta. On the other hand, Moore-Penrose hermitian elements, that is $C^*$-algebra elements which coincide with their Moore-Penrose inverse, are introduced and studied. In fact,these elements will be fully characterized both in the Hilbert space and in the $C^*$-algebra setting. Furthermore, it will be proved that an element is normal and Moore-Penrose hermitian if and only if it is a hermitian partial isometry.