On the numerical ranges of matrix products

Let A and C be n × n complex matrices. The C -numerical range of A is defined as the set W C ( A ) = { tr ( C U ∗ A U ) : U ∈ M n , U ∗ U = I n } . We study classes of matrices that two matrices A , B in the respective class satisfy W C ( A B ) = W C ( B A ) for a certain complex matrix C .