On the maximal numerical range of the bimultiplication M2,A,B

Let B(H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H. For A, B is an element of B(H), define the bimultiplication operator M-2,M-A,M-B on the class of Hilbert-Schmidt operators by M-2,M-A,M-B(X) = AXB. In this paper, we show that if B is normal, then co(W-0(A)W-0(B)) subset of W-0(M-2,M-A,M-B), where co stands for the convex hull and W-0(.) denotes the maximal numerical range. If in addition, A is hyponormal, this inclusion becomes an equality. Some remarks about the maximal numerical range of the generalized derivation delta(2 A,B) on the class of Hilbert-Schmidt operators are also given.