ON SOME FUNCTIONAL EQUATION IN STANDARD OPERATOR ALGEBRAS

In this paper we prove the following result. Let n >= 3 be some fixed integer, let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators on X, and let A(X) subset of L(X) be a standard operator algebra. Suppose there exists a linear mapping D : A(X) -> L(X) satisfying the relation 2(n-2) D(A(n)) = Sigma(n-2)(i=0)((n-2)(i))A(i) D(A(2))A(n-2-i) + (2(n-2) - 1)(D(A)A(-1) + An(-1) D)) Sigma(n-2)(i=1) (Sigma(i)(k=2) (2(k-1) -1)((n-k-2)(i-k)) + Sigma(n-1-i)(k=2)(2(k-1) -1) ((n-k-2)(n-i-k-1))A(i)D(A)An(-1-i) for all A is an element of A(X). In this case D is of the form D(A) = [B,A] for all A is an element of A(X), and some fixed B is an element of L(X). In particular, D is continuous. This result is related to a classical result of Chernoff.