Jacobson’s lemma and Cline’s formula for weighted generalized inverses in a ring with involution

Abstract

Let R be a ring with involution and e, f ∈ R be Hermitian and invertible. We first present some equivalent conditions for paq to be {1, 3 f }-invertible, assuming that p, a, q ∈ R with p′pa = a = aqq′ for
some p′, q′ ∈ R and a is {1, 3e}-invertible. Then, these results are applied to give the sufficient and necessary conditions under which Jacobson’s lemma and Cline’s formula for weighted pseudo core inverses hold. Also, Jacobson’s lemma for weighted Moore-Penrose inverses is investigated