On the numerical ranges of matrix products

Let AA and CC be n×nn×n complex matrices. The CC-numerical range of AA is defined as the set WC(A)={tr(CU∗AU):U∈Mn,U∗U=In}. We study classes of matrices that two matrices AA, BB in the respective class satisfy WC(AB)=WC(BA)WC(AB)=WC(BA) for a certain complex matrix CC.