The λ-Aluthge transform of EP matrices

Abstract

In this work we present some relationships between an EP matrix T, its Aluthge transform ?(T) or the ?-Aluthge transform ??(T) and the Moore-Penrose inverse T+. We prove that the ?-Aluthge transform of T is also an EP matrix, and the same thing holds for ??(T)+ and ??(T+). Also, we explore the product ??(T)T, the connections between ?(T) and T+ as well as the reverse order law for generalized inverses which are associated with ??(T). Finally, it is verified that the ranges of T and ??(T) are equal in the case of EP matrices.