On elements whose Moore–Penrose inverse is idempotent in a ∗-ring

Abstract

: In this paper, we investigate the elements whose Moore–Penrose inverse is idempotent in a ∗-ring. Let Rbe a ∗-ring and a ∈ R†. Firstly, we give a concise characterization for the idempotency of a†as follows: a ∈ R†and a†is idempotent if and only if a ∈ R# and a2 = aa∗a, which connects Moore–Penrose invertibility and groupinvertibility. Secondly, we generalize the results of Baksalary and Trenkler from complex matrices to ∗-rings. Moreequivalent conditions which ensure the idempotency of a†are given. Particularly, we provide the characterizations forboth a and a† being idempotent. Finally, the equivalent conditions under which a is EP and a†is idempotent areinvestigated