Abstract:

For n-tuples A = (At, ..., AN) and B = (BI, ..., Bn) of operators on aHilber! space
H, let RA,B denote the operator on L(H) defined by RA,B(X) = ~i=1AiX Bi. In
this paper we prove that
co c~i13i : (a,,..., an) C W(A), (131, ..., 1~) e W(B) C Wo(RA,B)
where W is the joint spatial numerical range and W0 is the numerical range. We
will show also that this inclusion becomes an equality when I~A,B is taken to be
a generalized derivation, and it is strict when RA,B is taken to be an elementary
multiplication operator induced by non scalar self-adjoints operators.