The numerical range of elementary operators

For n-tuplesA=(A 1,...,A n ) andB=(B 1,...,B n ) of operators on a Hilbert spaceH, letR A,B denote the operator onL(H) defined by RA,B(X)=(sigma) A iXB i. In this paper we prove that co{(sigma) (alpha)i(beta)I : ((alpha)1,…,(alpha)n (element of) W(A),((beta)1,…,(beta)n) (element of) W(B)}^- (subset of) Wo(R A,B) whereW is the joint spatial numerical range andW 0 is the numerical range. We will show also that this inclusion becomes an equality whenR A,B is taken to be a generalized derivation, and it is strict whenR A,B is taken to be an elementary multiplication operator induced by non scalar self-adjoints operators.