ON SOME FUNCTIONAL EQUATIONS ON STANDARD OPERATOR ALGEBRAS

The main purpose of this paper is to prove the following result. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators on X, let Lambda(X) subset of L(X) be a standard operator algebra, and let T:A(X) -> L(X) be an additive mapping satisfying the relation T(A(2n+1)) -Sigma(2n+1)(i=1)(-1)(i+1)A(i-1)T(A)A(2n+1-i), for all A is an element of A(X) and some fixed integer n >= 1. In this case T is of the form T(A) = AB+BA, for all A is an element of A(X) and some fixed B is an element of L(X). In particular, T is continuous.