Pseudo core inverses of a sum of morphisms

Abstract

Let be an additive category with an involution ∗. Suppose that both ϕ : X → X is a morphism of with pseudo core inverse ϕ and η : X → X is a morphism of such that 1 + ϕ η is invertible. Let α = (1 + ϕ η) − 1 , β = (1+ηϕ ) − 1 , ε = (1−ϕϕ )ηα(1−ϕ ϕ), γ = α(1−ϕ ϕ)ηϕ β, σ = αϕ ϕα− 1(1− ϕϕ )β, δ = β∗ (ϕ ) ∗η∗ (1−ϕϕ )β. Then we present a sufficient condition such that f = ϕ + η − ε has pseudo core inverse and give the corresponding expression. Let R be a unital ∗-ring and J(R) its Jacobson radical. If a is pseudo core invertible with the pseudo core inverse a and j ∈ J(R), we also give a sufficient condition which ensures that a + j − ε has pseudo core inverse. Thus, these results generalize recent results on core inverse.