Double centralizing automorphisms on prime, semiprime rings and Banach algebras

Let R be a prime ring and L a non-central Lie ideal of R. Our aim in this paper is to classify automorphisms g of R satisfying the following algebraic identity: \(\begin{aligned} \big [[g(x),x],x\big ]\in Z(R) \text { for all } x\in L. \end{aligned}\) $$\begin{array}{c}\hfill [[g(x),x],x]\in Z(R)\text{for all}x\in L.\end{array}$$ Moreover, some investigations of the same identity on semiprime rings were provided. Finally, as an application, we also obtain some range inclusion results with automorphisms on noncommutative Banach algebras.