ON CERTAIN EQUATION RELATED TO DERIVATIONS ON STANDARD OPERATOR ALGEBRAS AND SEMIPRIME RINGS

In this paper we prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let A(X) be a standard operator algebra on X and let L (X) be an algebra of all bounded linear operators on X. Suppose we have a linear mapping D: A(X) -> L (X) satisfying the relation D(A(m+n))=D(A(m))(An)+A(m)D(A(n)) for all A is an element of A(X) and some fixed integers m >= 1,n >= 1. In this case there exists B is an element of (X), such that D(A)=AB - BA holds for all A is an element of F(X), where F (X) denotes the ideal of all finite rank operators in L (X). Besides, D(A(m))=A(m)B - BA(m) is fulfilled for all A is an element of A(X).