On some functional equations on standard operator algebras

摘要

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 A(X) L(X) be a standard operator algebra, and let T : A(X) → L(X) be an additive mapping satisfying the relation T(A2n+1) = ∑i=12n+1 (-1)i+1 Ai-1 T(A) A2n+1-i, for all A A(X) and some fixed integer n ≥ 1. In this case T is of the form T(A) = AB + BA, for all A A(X) and some fixed B L(X). In particular, T is continuous.