On star-centers of some generalized numerical ranges and diagonals of normal matrices

For any n×n matrices A and C, we consider the star-centers of three sets, namely, the C-numerical range WC(A) of A, the set diag U(A) of diagonals of matrices in the unitary orbit of A, and the set S(A) of matrices whose C-numerical ranges are contained in WC(A) for all C. For normal matrices A, we show that the set of star-centers of WA*(A) is a bounded closed real interval, and give complete description of the sets of star-centers of diag U(A) and of S(A). In particular, we show that if A is normal with noncollinear eigenvalues, then each of S(A) and diag U(A) has exactly one star-center. For general square matrices A, we also give sufficient conditions for the sets of star-centers of diag U(A) and of S(A) to be singleton sets.