Characterizations of weighted core inverses in rings with involution

Let R be a unital ring with involution. The author investigates the characterizations and representations of weighted core inverse of an element in R by idempotents and units. Let a is an element of R, e is an element of R be an invertible Hermitian element and n >= 1. We prove that a is e-core invertible if and only if there exists an element (or an idempotent) p such that (ep)* = ep, pa = 0 and a(n) +p (or a'(1 - p) + p) is invertible. As a consequence, for two invertible Hermitian elements e and f in R, a is weighted-EP with respect to (e, f) if and only if there exists an element (or an idempotent) p such that (ep)* = ep, (fp)* = fp, pa = ap = 0 and a(n) + p (or a(n) (1 - p) + p) is invertible. These results generalize and improve conclusions in [T. T. Li and J. L. Chen, Characterizations of core and dual core inverses in rings with involution, Linear Multilinear Algebra 66 (2018) 717-730].