Moore-Penrose inverse and partial orders on Hilbert space operators

Abstract

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 operator orders. Finally, we study the operator B†A† as different weighted inverses of AB.