Developing Reverse Order Law for the Moore–Penrose Inverse with the Product of Three Linear Operators

In this paper, we study the reverse order law for the Moore–Penrose inverse of the product of three bounded linear operators in Hilbert spaces. We first present some equivalent conditions for the existence of the reverse order law A B C † = C † B † A † . Moreover, several equivalent statements of ℛ A A ∗ A B C = ℛ A B C and ℛ C ∗ C A B C ∗ = ℛ A B C ∗ are also deducted by the theory of operators.

Triple reverse order law of Drazin invertible operators

In this paper we study the triple reverse-order law (ABC)D = CDBDAD for the Drazin invertible operators A,B and C under the commutative relations [AB, B] = 0, [BC, B] = 0 and [AB, BC] = 0.

Generalized Inverses

In this survey paper we present some aspects of generalized inverses, which are related to inner and outer invertibility, Moore-Penrose inverse, the appropriate reverse order law, and Drazin inverse.