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Letdenote the algebra of all bounded linear operators acting on a complex Hilbert space. For, define the bimultiplication operatoron the class of Hilbert–Schmidt operators by. In this paper, we show that if  B  is normal, thenwhere  co  stands for the convex hull anddenotes 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 derivationon the class of Hilbert–Schmidt operators are also given.

On the maximal numerical range of the bimultiplication M 2,A,B