a note on the maximal numerical range of the bimultiplication m 2,a,b

Let b(h) denote the algebra of all bounded linear operators acting on a complex hilbert space h. for a, b ∈ b(h), define the bimultiplication operator m2,a,b on the class of hilbert-schmidt operators by m2,a,b(x) = axb. in this paper, we show that if b ∗ , the adjoint operator of b, is hyponormal, then co w0(a)w0(b)
⊆ w0(m2,a,b), where co stands for the convex hull and w0(.) denotes the maximal numerical range. if in addition, a is hyponormal, we show that co w0(a)w0(b)
= w0(m2,a,b