Characterization of ($\alpha$,$\alpha$)-derivation on $B(X)$

Abstract

Let $X$ be a Banach algebra and $B(X)$ be the set of all bounded linear operators on $X$. Suppose that $\alpha: B(X) \rightarrow B(X)$ is an automorphism. We say that a mapping $\delta$ from $B(X)$ into itself is derivable at $G \in B(X)$ if $\delta(G) = \alpha(A)\delta(B) + \delta(A)\alpha(B)$ for all $A, B \in B(X)$ with $AB = G$. We say that an element $G \in B(X)$ is an $(\alpha,\alpha)$-all derivable point of $B(X)$ if every $(\alpha,\alpha)$-derivable mapping $\delta$ at G is an $(\alpha,\alpha)$-derivation. In this paper, we show that every $(\alpha,\alpha)$-derivable mapping at a nonzero element in $B(X)$ is an $(\alpha,\alpha)$-derivation.