The purpose of this paper is to characterize several classes of functional identities involving inverses with related mappings from a unital Banach algebra $\mathcal{ A}$ over the complex field into a unital $\mathcal{ A}$-bimodule $\mathcal{ M}$. Let $N$ be a fixed invertible element in $\mathcal{ A}$, $M$ be a fixed element in $\mathcal{ M}$, and $n$ be a positive integer. We investigate the forms of additive mappings $f$, $g$ from $\mathcal{ A}$ into $\mathcal{ M}$ satisfying one of the following identities: \begin{ equation*} \begin{ aligned} &f(A)A- Ag(A) = 0\\ &f(A)+ g(B)\star A= M\\ &f(A)+A^{ n}g(A^{ -1})=0\\ &f(A)+A^{ n}g(B)=M \end{ aligned} \qquad \begin{ aligned} &\text{ for each invertible element}~A\in\mathcal{ A}; \\ &\text{ whenever}~ A,B\in\mathcal{ A}~\text{ with}~AB=N;\\ &\text{ for each invertible element}~A\in\mathcal{ A}; \\ &\text{ whenever}~ A,B\in\mathcal{ A}~\text{ with}~AB=N, \end{ aligned} \end{ equation*} where $\star$ is either the Jordan product $A\star B = AB+BA$ or the Lie product $A\star B = AB-BA$.
Categories: math.FA math.OA