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 MoorePenrose 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.