标题： 涉及逆元的巴拿赫代数上的函数恒等式 显示英文标题

标题： Functional identities involving inverses on Banach algebras

摘要： 本文的目的是表征涉及逆元及相关映射的若干函数恒等式，这些恒等式从复数域上的单位Banach代数$\mathcal{A}$映射到一个单位$\mathcal{A}$-双模$\mathcal{M}$。 设$N$为$\mathcal{A}$中的一个固定可逆元素，$M$为$\mathcal{M}$中的一个固定元素，$n$为一个正整数。 我们研究从$\mathcal{A}$到$\mathcal{M}$的加法映射$f$,$g$的形式，满足以下恒等式之一：\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*}，其中$\star$是 Jordan 乘积$A\star B = AB+BA$或者李乘积$A\star B = AB-BA$。 显示英文摘要

摘要： 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 $A\star B = AB-BA$.