Weighted w-core inverses in rings

Abstract

Let R be a unital *-ring. For any a, s, t, v, w ∈ R we define the weighted w-core inverse and the weighted dual s-core inverse, extending the w-core inverse and the dual s-core inverse, respectively. An element a ∈ R has a weighted w-core inverse with the weight v if there exists some x ∈ R such that awxvx = x, xvawa = a and (awx)* = awx. Dually, an element a ∈ R has a weighted dual s-core inverse with the weight t if there exists some y ∈ R such that ytysa = y, asaty = a and (ysa)* = ysa. Several characterizations of weighted w-core invertible and weighted dual s-core invertible elements are given when weights v and t are invertible Hermitian elements. Also, the relations among the weighted w-core inverse, the weighted dual s-core inverse, the e-core inverse, the dual f-core inverse, the weighted Moore-Penrose inverse and the (v, w)-(b, c)-inverse are considered.