Maximal numerical range of tensor product of two operators

Let A and B be two operators acting on two complex Hilbert spaces $ \mathscr {H} $ H and $ \mathscr {K} $ K, respectively. In this paper, we show under some hyponormality conditions, the following equality $$\begin{align*} W_{0}(A\otimes B) = \mathrm{co}\left( W_{0}(A) \cdot W_{0}(B) \right) \end{align*}$$W0(A circle times B)=co(W0(A)& sdot;W0(B)) holds, where $ A\otimes B $ A circle times B, $ W_{0}(\cdot ) $ W0(& sdot;) and $ \mathrm {co}(\cdot ) $ co(& sdot;) denote respectively the tensor product of A and B, the maximal numerical range and convex hull. Furthermore, we provide a necessary and sufficient condition for the operator $ A\otimes B $ A circle times B being hyponormal.