On functional equations related to generalized inner derivations on standard operator algebras

Abstract

Our aim is to prove the following result. Let $X$ be a real or complex Banach space, let $\mathcal{ℒ}\left(X\right)$ be the algebra of all bounded linear operators on $X$ and let $\mathcal{𝒜}\left(X\right)\subseteq \mathcal{ℒ}\left(X\right)$ be a standard operator algebra, which possesses the identity operator. Suppose there exists a linear mapping $F:\mathcal{𝒜}\left(X\right)\to \mathcal{ℒ}\left(X\right)$ satisfying the relation $F\left(A^{n+2}\right)={\sum }_{i=1}^{n+1}{A}^{i-1}F\left(A^2\right)A^{n+1-i}-{\sum }_{i=1}^n{A}^{i}F\left(A\right)A^{n+1-i}$ for all $A\in \mathcal{𝒜}\left(X\right)$ and some fixed integer $n\ge 1$. In this case $F$ is of the form $F\left(A\right)=A{B}_1+B_{2}A$ for all $A\in \mathcal{𝒜}\left(X\right)$ and some fixed $B_1,B_2\in \mathcal{ℒ}\left(X\right)$. In particular, $F$ is continuous.