A note on th maximal numerical range of the bimultiplication M_2,A,B

Abstract

Let $\mathcal B(H)$ denote the algebra of all bounded linear operators acting on a complex Hilbert space $ H$. For $A,B\in \mathcal B(H)$, define the bimultiplication operator $M_{2,A,B}$ on the class of Hilbert-Schmidt operators by $M_{2,A,B}(X)= AXB$. In this paper, we show that if $B^{*}$, the adjoint operator of $B$, is hyponormal, then $$ co(W_{0}(A) W_{0}(B))\subseteq W_{0}(M_{2,A,B}),$$ where $co$ stands for the convex hull and $W_{0}(.)$ denotes the maximal numerical range. If in addition, $A$ is hyponormal, we show that $$ co(W_{0}(A) W_{0}(B))= W_{0}(M_{2,A,B}).$$