Involution on prime rings with endomorphisms

Let $\mathcal{R}$ be a prime ring with involution $'*'$ and $\psi: \mathcal{R} \rightarrow \mathcal{R}$ be an endomorphism on $\mathcal{R}$. In this article, we study the action of involution $'*', $ and the effect of endomorphism $\psi$ satisfying $[\psi(x), \psi(x^*)]-[x, x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$. In particular, we prove that any centralizing involution on prime rings with involution of characteristic different from two is of the first kind or $\mathcal{R}$ satisfies $s_4$, the standard polynomial identity in four variables. Further, we establish that if a prime ring $\mathcal{R}$ with involution of characteristic different from two admits a non-trivial endomorphism $\psi$ such that $[\psi(x), \psi(x^*)]-[x, x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$, then the involution is of the first kind or $\mathcal{R}$ satisfies $s_4$ and $[\psi(x), x] = 0$ for all $x\in \mathcal{R}$.