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: [[g(x),x],x]is an element of Z(R)for allx is an element of L.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big [[g(x),x],x\big ]\in Z(R) \text { for all } x\in L. \end{aligned}$$\end{document}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.