Core and Dual Core Inverses of a Sum of Morphisms

Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $\varphi : X \rightarrow X$ is a morphism of $\mathscr{C}$ with core inverse $\varphi^{\co} : X \rightarrow X$ and $\eta : X \rightarrow X$ is a morphism of $\mathscr{C}$ such that $1_X+\varphi^{\co}\eta$ is invertible. Let $\alpha=(1_X+\varphi^{\co}\eta)^{-1},$ $\beta=(1_X+\eta\varphi^{\co})^{-1},$ $\varepsilon=(1_X-\varphi\varphi^{\co})\eta\alpha(1_X-\varphi^{\co}\varphi),$ $\gamma=\alpha(1_X-\varphi^{\co}\varphi)\beta^{-1}\varphi\varphi^{\co}\beta,$ $\sigma=\alpha\varphi^{\co}\varphi\alpha^{-1}(1_X-\varphi\varphi^{\co})\beta,$ $\delta=\beta^{\ast}(\varphi^{\co})^{\ast}\eta^{\ast}(1_X-\varphi\varphi^{\co})\beta.$ Then $f=\varphi+\eta-\varepsilon$ has a core inverse if and only if $1_X-\gamma$, $1_X-\sigma$ and $1_X-\delta$ are invertible. Moreover, the expression of the core inverse of $f$ is presented. Let $R$ be a unital $\ast$-ring and $J(R)$ its Jacobson radical, if $a\in R^{\co}$ with core inverse $a^{\co}$ and $j\in J(R)$, then $a+j\in R^{\co}$ if and only if $(1-aa^{\co})j(1+a^{\co}j)^{-1}(1-a^{\co}a)=0$. We also give the similar results for the dual core inverse.