IDENTITIES RELATED TO DERIVATIONS AND CENTRALIZERS ON STANDARD OPERATOR ALGEBRAS

Abstract

In this paper identities related to derivations and centralizers on operator algebras are considered. We prove the following result which is related to a classical result of Chernoff. Let $X$ be a real or complex Banach space, let $L(X)$ and $F(X)$ be the algebra of all bounded linear operators and the ideal of all finite rank operators on $X$, respectively. Suppose there exist linear mappings $D,G : F(X) \to L(X)$ such that $D(A^2) = D(A)A + AG(A)$ and $G(A^2) = G(A)A + AD(A)$ is fulfilled for all $A \in F(X)$. In this case there exists $B \in L(X)$ such that $D(A) = G(A) = [A,B]$ holds for all $A \in F(X)$.