Flat portions on the boundary of the Davis–Wielandt shell of 3-by-3 matrices

Let TT be an n×nn×n matrix. The Davis–Wielandt shell of TT is defined as the setW(T,T∗T)={(x∗Tx,x∗T∗Tx):x∈C×R,x∗x=1}.In this paper, we discuss the existence of flat portions on the boundary of the Davis–Wielandt shell of 3-by-3 matrices.